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L3 Lab 3

case inr.inr z : ℂ hre : z.re < 0 him : z.im < 0 ⊢ -(π / 2) ≤ arg z ↔ 0 ≤ z.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) ·

simp only [him.not_le]

theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5

theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inr z : ℂ hre : z.re < 0 him : z.im < 0 ⊢ -(π / 2) ≤ arg z ↔ False

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le]

rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs]

theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5

theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inr z : ℂ hre : z.re < 0 him : z.im < 0 ⊢ z.re ≠ 0 case inr.inr z : ℂ hre : z.re < 0 him : z.im < 0 ⊢ 0 < abs z

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs]

exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne]

theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.349_0.CflASCTDE9UCom5

theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ ⊢ -(π / 2) < arg z ↔ 0 < z.re ∨ 0 ≤ z.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by

rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff]

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.364_0.CflASCTDE9UCom5

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ ⊢ (0 ≤ z.re ∨ 0 ≤ z.im) ∧ ¬(z.re = 0 ∧ z.im < 0) ↔ 0 < z.re ∨ 0 ≤ z.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff]

rcases lt_trichotomy z.re 0 with hre | hre | hre

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.364_0.CflASCTDE9UCom5

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inl z : ℂ hre : z.re < 0 ⊢ (0 ≤ z.re ∨ 0 ≤ z.im) ∧ ¬(z.re = 0 ∧ z.im < 0) ↔ 0 < z.re ∨ 0 ≤ z.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre ·

simp [hre.ne, hre.not_le, hre.not_lt]

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.364_0.CflASCTDE9UCom5

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl z : ℂ hre : z.re = 0 ⊢ (0 ≤ z.re ∨ 0 ≤ z.im) ∧ ¬(z.re = 0 ∧ z.im < 0) ↔ 0 < z.re ∨ 0 ≤ z.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] ·

simp [hre]

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.364_0.CflASCTDE9UCom5

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inr z : ℂ hre : 0 < z.re ⊢ (0 ≤ z.re ∨ 0 ≤ z.im) ∧ ¬(z.re = 0 ∧ z.im < 0) ↔ 0 < z.re ∨ 0 ≤ z.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] ·

simp [hre, hre.le, hre.ne']

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.364_0.CflASCTDE9UCom5

lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ ⊢ |arg z| ≤ π / 2 ↔ 0 ≤ z.re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by

rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff]

@[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.371_0.CflASCTDE9UCom5

@[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ ⊢ ↑(arg ((starRingEnd ℂ) x)) = -↑(arg x)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by

by_cases h : arg x = π

@[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.377_0.CflASCTDE9UCom5

@[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case pos x : ℂ h : arg x = π ⊢ ↑(arg ((starRingEnd ℂ) x)) = -↑(arg x)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

simp [arg_conj, h]

@[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

Mathlib.Analysis.SpecialFunctions.Complex.Arg.377_0.CflASCTDE9UCom5

@[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case neg x : ℂ h : ¬arg x = π ⊢ ↑(arg ((starRingEnd ℂ) x)) = -↑(arg x)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

simp [arg_conj, h]

@[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

Mathlib.Analysis.SpecialFunctions.Complex.Arg.377_0.CflASCTDE9UCom5

@[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ ⊢ ↑(arg x⁻¹) = -↑(arg x)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by

by_cases h : arg x = π

@[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.382_0.CflASCTDE9UCom5

@[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case pos x : ℂ h : arg x = π ⊢ ↑(arg x⁻¹) = -↑(arg x)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

simp [arg_inv, h]

@[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

Mathlib.Analysis.SpecialFunctions.Complex.Arg.382_0.CflASCTDE9UCom5

@[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case neg x : ℂ h : ¬arg x = π ⊢ ↑(arg x⁻¹) = -↑(arg x)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

simp [arg_inv, h]

@[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;>

Mathlib.Analysis.SpecialFunctions.Complex.Arg.382_0.CflASCTDE9UCom5

@[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ hi : 0 < x.im ⊢ arg (-x) = arg x - π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by

rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)]

theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.387_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ hi : 0 < x.im ⊢ -arccos ((-x).re / abs (-x)) = arccos (x.re / abs x) - π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)]

simp [neg_div, Real.arccos_neg]

theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.387_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ hi : x.im < 0 ⊢ arg (-x) = arg x + π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by

rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)]

theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.392_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ hi : x.im < 0 ⊢ arccos ((-x).re / abs (-x)) = -arccos (x.re / abs x) + π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)]

simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg]

theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.392_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ ⊢ arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by

rcases lt_trichotomy x.im 0 with (hi | hi | hi)

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inl x : ℂ hi : x.im < 0 ⊢ arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) ·

simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl x : ℂ hi : x.im = 0 ⊢ arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] ·

rw [(ext rfl hi : x = x.re)]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl x : ℂ hi : x.im = 0 ⊢ arg (-↑x.re) = arg ↑x.re - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)]

rcases lt_trichotomy x.re 0 with (hr | hr | hr)

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inl x : ℂ hi : x.im = 0 hr : x.re < 0 ⊢ arg (-↑x.re) = arg ↑x.re - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) ·

rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inl x : ℂ hi : x.im = 0 hr : x.re < 0 ⊢ 0 = π - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le]

simp [hr]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inl x : ℂ hi : x.im = 0 hr : x.re = 0 ⊢ arg (-↑x.re) = arg ↑x.re - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] ·

simp [hr, hi, Real.pi_ne_zero]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inr x : ℂ hi : x.im = 0 hr : 0 < x.re ⊢ arg (-↑x.re) = arg ↑x.re - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] ·

rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inr x : ℂ hi : x.im = 0 hr : 0 < x.re ⊢ π = 0 - π ↔ 0 < (↑x.re).im ∨ (↑x.re).im = 0 ∧ (↑x.re).re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]

simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inr x : ℂ hi : 0 < x.im ⊢ arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] ·

simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos]

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.397_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ ⊢ arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by

rcases lt_trichotomy x.im 0 with (hi | hi | hi)

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inl x : ℂ hi : x.im < 0 ⊢ arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) ·

simp [hi, arg_neg_eq_arg_add_pi_of_im_neg]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl x : ℂ hi : x.im = 0 ⊢ arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] ·

rw [(ext rfl hi : x = x.re)]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl x : ℂ hi : x.im = 0 ⊢ arg (-↑x.re) = arg ↑x.re + π ↔ (↑x.re).im < 0 ∨ (↑x.re).im = 0 ∧ 0 < (↑x.re).re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)]

rcases lt_trichotomy x.re 0 with (hr | hr | hr)

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inl x : ℂ hi : x.im = 0 hr : x.re < 0 ⊢ arg (-↑x.re) = arg ↑x.re + π ↔ (↑x.re).im < 0 ∨ (↑x.re).im = 0 ∧ 0 < (↑x.re).re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) ·

rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inl x : ℂ hi : x.im = 0 hr : x.re < 0 ⊢ 0 = π + π ↔ (↑x.re).im < 0 ∨ (↑x.re).im = 0 ∧ 0 < (↑x.re).re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le]

simp [hr.not_lt, ← two_mul, Real.pi_ne_zero]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inl x : ℂ hi : x.im = 0 hr : x.re = 0 ⊢ arg (-↑x.re) = arg ↑x.re + π ↔ (↑x.re).im < 0 ∨ (↑x.re).im = 0 ∧ 0 < (↑x.re).re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] ·

simp [hr, hi, Real.pi_ne_zero.symm]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inr x : ℂ hi : x.im = 0 hr : 0 < x.re ⊢ arg (-↑x.re) = arg ↑x.re + π ↔ (↑x.re).im < 0 ∨ (↑x.re).im = 0 ∧ 0 < (↑x.re).re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] ·

rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inr x : ℂ hi : x.im = 0 hr : 0 < x.re ⊢ π = 0 + π ↔ (↑x.re).im < 0 ∨ (↑x.re).im = 0 ∧ 0 < (↑x.re).re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]

simp [hr]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inr x : ℂ hi : 0 < x.im ⊢ arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] ·

simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero]

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.412_0.CflASCTDE9UCom5

theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x : ℂ hx : x ≠ 0 ⊢ ↑(arg (-x)) = ↑(arg x) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by

rcases lt_trichotomy x.im 0 with (hi | hi | hi)

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inl x : ℂ hx : x ≠ 0 hi : x.im < 0 ⊢ ↑(arg (-x)) = ↑(arg x) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) ·

rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add]

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl x : ℂ hx : x ≠ 0 hi : x.im = 0 ⊢ ↑(arg (-x)) = ↑(arg x) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] ·

rw [(ext rfl hi : x = x.re)]

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl x : ℂ hx : x ≠ 0 hi : x.im = 0 ⊢ ↑(arg (-↑x.re)) = ↑(arg ↑x.re) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)]

rcases lt_trichotomy x.re 0 with (hr | hr | hr)

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inl x : ℂ hx : x ≠ 0 hi : x.im = 0 hr : x.re < 0 ⊢ ↑(arg (-↑x.re)) = ↑(arg ↑x.re) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) ·

rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero]

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inl x : ℂ hx : x ≠ 0 hi : x.im = 0 hr : x.re = 0 ⊢ ↑(arg (-↑x.re)) = ↑(arg ↑x.re) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] ·

exact False.elim (hx (ext hr hi))

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inl.inr.inr x : ℂ hx : x ≠ 0 hi : x.im = 0 hr : 0 < x.re ⊢ ↑(arg (-↑x.re)) = ↑(arg ↑x.re) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) ·

rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add]

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inr.inr x : ℂ hx : x ≠ 0 hi : 0 < x.im ⊢ ↑(arg (-x)) = ↑(arg x) + ↑π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] ·

rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi]

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.427_0.CflASCTDE9UCom5

theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

r : ℝ hr : 0 < r θ : ℝ ⊢ arg (↑r * (cos ↑θ + sin ↑θ * I)) = toIocMod two_pi_pos (-π) θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by

have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.440_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

r : ℝ hr : 0 < r θ : ℝ ⊢ toIocMod two_pi_pos (-π) θ ∈ Set.Ioc (-π) π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by

convert toIocMod_mem_Ioc _ _ θ

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.440_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h.e'_5.h.e'_4 r : ℝ hr : 0 < r θ : ℝ ⊢ π = -π + 2 * π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ

ring

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ

Mathlib.Analysis.SpecialFunctions.Complex.Arg.440_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

r : ℝ hr : 0 < r θ : ℝ hi : toIocMod two_pi_pos (-π) θ ∈ Set.Ioc (-π) π ⊢ arg (↑r * (cos ↑θ + sin ↑θ * I)) = toIocMod two_pi_pos (-π) θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring

convert arg_mul_cos_add_sin_mul_I hr hi using 3

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring

Mathlib.Analysis.SpecialFunctions.Complex.Arg.440_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h.e'_2.h.e'_1.h.e'_6 r : ℝ hr : 0 < r θ : ℝ hi : toIocMod two_pi_pos (-π) θ ∈ Set.Ioc (-π) π ⊢ cos ↑θ + sin ↑θ * I = cos ↑(toIocMod two_pi_pos (-π) θ) + sin ↑(toIocMod two_pi_pos (-π) θ) * I

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3

simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi]

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3

Mathlib.Analysis.SpecialFunctions.Complex.Arg.440_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

θ : ℝ ⊢ arg (cos ↑θ + sin ↑θ * I) = toIocMod two_pi_pos (-π) θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by

rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one]

theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.450_0.CflASCTDE9UCom5

theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

r : ℝ hr : 0 < r θ : ℝ ⊢ arg (↑r * (cos ↑θ + sin ↑θ * I)) - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by

rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul]

theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.456_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋

Mathlib_Analysis_SpecialFunctions_Complex_Arg

r : ℝ hr : 0 < r θ : ℝ ⊢ ↑(- -⌊(-π + 2 * π - θ) / (2 * π)⌋) * (2 * π) = 2 * π * ↑⌊(π - θ) / (2 * π)⌋

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul]

ring_nf

theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.456_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋

Mathlib_Analysis_SpecialFunctions_Complex_Arg

θ : ℝ ⊢ arg (cos ↑θ + sin ↑θ * I) - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by

rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one]

theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.464_0.CflASCTDE9UCom5

theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋

Mathlib_Analysis_SpecialFunctions_Complex_Arg

r : ℝ hr : 0 < r θ : Angle ⊢ ↑(arg (↑r * (↑(Angle.cos θ) + ↑(Angle.sin θ) * I))) = θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by

induction' θ using Real.Angle.induction_on with θ

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.470_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h r : ℝ hr : 0 < r θ : ℝ ⊢ ↑(arg (↑r * (↑(Angle.cos ↑θ) + ↑(Angle.sin ↑θ) * I))) = ↑θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ

rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub]

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ

Mathlib.Analysis.SpecialFunctions.Complex.Arg.470_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h r : ℝ hr : 0 < r θ : ℝ ⊢ ∃ k, arg (↑r * (↑(Real.cos θ) + ↑(Real.sin θ) * I)) - θ = 2 * π * ↑k

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub]

use ⌊(π - θ) / (2 * π)⌋

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.470_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h r : ℝ hr : 0 < r θ : ℝ ⊢ arg (↑r * (↑(Real.cos θ) + ↑(Real.sin θ) * I)) - θ = 2 * π * ↑⌊(π - θ) / (2 * π)⌋

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋

exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋

Mathlib.Analysis.SpecialFunctions.Complex.Arg.470_0.CflASCTDE9UCom5

theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

θ : Angle ⊢ ↑(arg (↑(Angle.cos θ) + ↑(Angle.sin θ) * I)) = θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by

rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one]

theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.479_0.CflASCTDE9UCom5

theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x y : ℂ hx : x ≠ 0 hy : y ≠ 0 ⊢ ↑(arg (x * y)) = ↑(arg x) + ↑(arg y)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by

convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3

theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.485_0.CflASCTDE9UCom5

theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h.e'_2.h.e'_1.h.e'_1 x y : ℂ hx : x ≠ 0 hy : y ≠ 0 ⊢ x * y = ↑(abs x * abs y) * (↑(Angle.cos (↑(arg x) + ↑(arg y))) + ↑(Angle.sin (↑(arg x) + ↑(arg y))) * I)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3

simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul]

theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3

Mathlib.Analysis.SpecialFunctions.Complex.Arg.485_0.CflASCTDE9UCom5

theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h.e'_2.h.e'_1.h.e'_1 x y : ℂ hx : x ≠ 0 hy : y ≠ 0 ⊢ x * y = ↑(abs x) * ↑(abs y) * (cexp (↑(arg x) * I) * cexp (↑(arg y) * I))

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul]

rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I]

theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.485_0.CflASCTDE9UCom5

theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x y : ℂ hx : x ≠ 0 hy : y ≠ 0 ⊢ ↑(arg (x / y)) = ↑(arg x) - ↑(arg y)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by

rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg]

theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.496_0.CflASCTDE9UCom5

theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ ⊢ Angle.toReal ↑(arg z) = arg z

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by

rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc]

@[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.501_0.CflASCTDE9UCom5

@[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ ⊢ arg z ∈ Set.Ioc (-π) π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc]

exact arg_mem_Ioc _

@[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.501_0.CflASCTDE9UCom5

@[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ θ : Angle ⊢ ↑(arg z) = θ ↔ arg z = Angle.toReal θ

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by

rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg]

theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.507_0.CflASCTDE9UCom5

theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x y : ℂ ⊢ ↑(arg x) = ↑(arg y) ↔ arg x = arg y

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by

simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg]

@[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.512_0.CflASCTDE9UCom5

@[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x y : ℂ hx₀ : x ≠ 0 hy₀ : y ≠ 0 ⊢ arg (x * y) = arg x + arg y ↔ arg x + arg y ∈ Set.Ioc (-π) π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by

rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc]

lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.517_0.CflASCTDE9UCom5

lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ hx_re : x.re < 0 hx_im : 0 < x.im ⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / abs x) + π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by

suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.532_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ hx_re : x.re < 0 hx_im : 0 < x.im h_forall_nhds : ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im ⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / abs x) + π case h_forall_nhds x z : ℂ hx_re : x.re < 0 hx_im : 0 < x.im ⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im

exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im

Mathlib.Analysis.SpecialFunctions.Complex.Arg.532_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h_forall_nhds x z : ℂ hx_re : x.re < 0 hx_im : 0 < x.im ⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ 0 < y.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le

refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im)

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le

Mathlib.Analysis.SpecialFunctions.Complex.Arg.532_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h_forall_nhds x z : ℂ hx_re : x.re < 0 hx_im : 0 < x.im ⊢ IsOpen fun y => y.re < 0 ∧ 0 < y.im

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im)

exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im)

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im)

Mathlib.Analysis.SpecialFunctions.Complex.Arg.532_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ hx_re : x.re < 0 hx_im : x.im < 0 ⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / abs x) - π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by

suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.541_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ hx_re : x.re < 0 hx_im : x.im < 0 h_forall_nhds : ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ y.im < 0 ⊢ arg =ᶠ[𝓝 x] fun x => arcsin ((-x).im / abs x) - π case h_forall_nhds x z : ℂ hx_re : x.re < 0 hx_im : x.im < 0 ⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ y.im < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0

exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0

Mathlib.Analysis.SpecialFunctions.Complex.Arg.541_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h_forall_nhds x z : ℂ hx_re : x.re < 0 hx_im : x.im < 0 ⊢ ∀ᶠ (y : ℂ) in 𝓝 x, y.re < 0 ∧ y.im < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2

refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0)

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2

Mathlib.Analysis.SpecialFunctions.Complex.Arg.541_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h_forall_nhds x z : ℂ hx_re : x.re < 0 hx_im : x.im < 0 ⊢ IsOpen fun y => y.re < 0 ∧ y.im < 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0)

exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero)

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0)

Mathlib.Analysis.SpecialFunctions.Complex.Arg.541_0.CflASCTDE9UCom5

theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ h : 0 < x.re ∨ x.im ≠ 0 ⊢ ContinuousAt arg x

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by

have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ h : 0 < x.re ∨ x.im ≠ 0 ⊢ abs x ≠ 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by

rw [abs.ne_zero_iff]

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ h : 0 < x.re ∨ x.im ≠ 0 ⊢ x ≠ 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff]

rintro rfl

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

z : ℂ h : 0 < 0.re ∨ 0.im ≠ 0 ⊢ False

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl

simp at h

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ h : 0 < x.re ∨ x.im ≠ 0 h₀ : abs x ≠ 0 ⊢ ContinuousAt arg x

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h

rw [← lt_or_lt_iff_ne] at h

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z : ℂ h : 0 < x.re ∨ x.im < 0 ∨ 0 < x.im h₀ : abs x ≠ 0 ⊢ ContinuousAt arg x

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h

rcases h with (hx_re | hx_im | hx_im)

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case inl x z : ℂ h₀ : abs x ≠ 0 hx_re : 0 < x.re ⊢ ContinuousAt arg x case inr.inl x z : ℂ h₀ : abs x ≠ 0 hx_im : x.im < 0 ⊢ ContinuousAt arg x case inr.inr x z : ℂ h₀ : abs x ≠ 0 hx_im : 0 < x.im ⊢ ContinuousAt arg x

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im)

exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm]

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im)

Mathlib.Analysis.SpecialFunctions.Complex.Arg.558_0.CflASCTDE9UCom5

theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 ⊢ Tendsto arg (𝓝[{z | z.im < 0}] z) (𝓝 (-π))

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by

suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 H : Tendsto (fun x => arcsin ((-x).im / abs x) - π) (𝓝[{z | z.im < 0}] z) (𝓝 (-π)) ⊢ Tendsto arg (𝓝[{z | z.im < 0}] z) (𝓝 (-π))

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) ·

refine' H.congr' _

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 H : Tendsto (fun x => arcsin ((-x).im / abs x) - π) (𝓝[{z | z.im < 0}] z) (𝓝 (-π)) ⊢ (fun x => arcsin ((-x).im / abs x) - π) =ᶠ[𝓝[{z | z.im < 0}] z] arg

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _

have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre)

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 H : Tendsto (fun x => arcsin ((-x).im / abs x) - π) (𝓝[{z | z.im < 0}] z) (𝓝 (-π)) this : ∀ᶠ (x : ℂ) in 𝓝 z, x.re < 0 ⊢ (fun x => arcsin ((-x).im / abs x) - π) =ᶠ[𝓝[{z | z.im < 0}] z] arg

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set

filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h x z✝ z : ℂ hre✝ : z.re < 0 him✝ : z.im = 0 H : Tendsto (fun x => arcsin ((-x).im / abs x) - π) (𝓝[{z | z.im < 0}] z) (𝓝 (-π)) this : ∀ᶠ (x : ℂ) in 𝓝 z, x.re < 0 a✝ : ℂ him : a✝.im < 0 hre : a✝.re < 0 ⊢ arcsin ((-a✝).im / abs a✝) - π = arg a✝

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre

rw [arg, if_neg hre.not_le, if_neg him.not_le]

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case H x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 ⊢ Tendsto (fun x => arcsin ((-x).im / abs x) - π) (𝓝[{z | z.im < 0}] z) (𝓝 (-π))

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le]

convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h.e'_5 x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 ⊢ 𝓝 (-π) = 𝓝 ((arcsin ∘ (im ∘ Neg.neg / ⇑abs)) z - π)

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 ·

simp [him]

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case H x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 ⊢ abs z ≠ 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] ·

lift z to ℝ using him

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case H.intro x z✝ : ℂ z : ℝ hre : (↑z).re < 0 ⊢ abs ↑z ≠ 0

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him

simpa using hre.ne

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him

Mathlib.Analysis.SpecialFunctions.Complex.Arg.576_0.CflASCTDE9UCom5

theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π))

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 ⊢ ContinuousWithinAt arg {z | 0 ≤ z.im} z

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him simpa using hre.ne #align complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by

have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_pos him]

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 ⊢ arg =ᶠ[𝓝[{z | 0 ≤ z.im}] z] fun x => arcsin ((-x).im / abs x) + π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him simpa using hre.ne #align complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by

have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre)

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by

Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 this : ∀ᶠ (x : ℂ) in 𝓝 z, x.re < 0 ⊢ arg =ᶠ[𝓝[{z | 0 ≤ z.im}] z] fun x => arcsin ((-x).im / abs x) + π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him simpa using hre.ne #align complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre)

filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre)

Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case h x z✝ z : ℂ hre✝ : z.re < 0 him✝ : z.im = 0 this : ∀ᶠ (x : ℂ) in 𝓝 z, x.re < 0 a✝ : ℂ him : 0 ≤ a✝.im hre : a✝.re < 0 ⊢ arg a✝ = arcsin ((-a✝).im / abs a✝) + π

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him simpa using hre.ne #align complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre

rw [arg, if_neg hre.not_le, if_pos him]

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre

Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 this : arg =ᶠ[𝓝[{z | 0 ≤ z.im}] z] fun x => arcsin ((-x).im / abs x) + π ⊢ ContinuousWithinAt arg {z | 0 ≤ z.im} z

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him simpa using hre.ne #align complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_pos him]

refine' ContinuousWithinAt.congr_of_eventuallyEq _ this _

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_pos him]

Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z

Mathlib_Analysis_SpecialFunctions_Complex_Arg

case refine'_1 x z✝ z : ℂ hre : z.re < 0 him : z.im = 0 this : arg =ᶠ[𝓝[{z | 0 ≤ z.im}] z] fun x => arcsin ((-x).im / abs x) + π ⊢ ContinuousWithinAt (fun x => arcsin ((-x).im / abs x) + π) {z | 0 ≤ z.im} z

/- Copyright (c) 2018 Chris Hughes. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson -/ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse #align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1" /-! # The argument of a complex number. We define `arg : ℂ → ℝ`, returning a real number in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, while `arg 0` defaults to `0` -/ noncomputable section namespace Complex open ComplexConjugate Real Topology open Filter Set /-- `arg` returns values in the range (-π, π], such that for `x ≠ 0`, `sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`, `arg 0` defaults to `0` -/ noncomputable def arg (x : ℂ) : ℝ := if 0 ≤ x.re then Real.arcsin (x.im / abs x) else if 0 ≤ x.im then Real.arcsin ((-x).im / abs x) + π else Real.arcsin ((-x).im / abs x) - π #align complex.arg Complex.arg theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by unfold arg; split_ifs <;> simp [sub_eq_add_neg, arg, Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2, Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg] #align complex.sin_arg Complex.sin_arg theorem cos_arg {x : ℂ} (hx : x ≠ 0) : Real.cos (arg x) = x.re / abs x := by rw [arg] split_ifs with h₁ h₂ · rw [Real.cos_arcsin] field_simp [Real.sqrt_sq, (abs.pos hx).le, *] · rw [Real.cos_add_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] · rw [Real.cos_sub_pi, Real.cos_arcsin] field_simp [Real.sqrt_div (sq_nonneg _), Real.sqrt_sq_eq_abs, _root_.abs_of_neg (not_le.1 h₁), *] #align complex.cos_arg Complex.cos_arg @[simp] theorem abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x := by rcases eq_or_ne x 0 with (rfl | hx) · simp · have : abs x ≠ 0 := abs.ne_zero hx apply Complex.ext <;> field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] set_option linter.uppercaseLean3 false in #align complex.abs_mul_exp_arg_mul_I Complex.abs_mul_exp_arg_mul_I @[simp] theorem abs_mul_cos_add_sin_mul_I (x : ℂ) : (abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x := by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] set_option linter.uppercaseLean3 false in #align complex.abs_mul_cos_add_sin_mul_I Complex.abs_mul_cos_add_sin_mul_I @[simp] lemma abs_mul_cos_arg (x : ℂ) : abs x * Real.cos (arg x) = x.re := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg re (abs_mul_cos_add_sin_mul_I x) @[simp] lemma abs_mul_sin_arg (x : ℂ) : abs x * Real.sin (arg x) = x.im := by simpa [-abs_mul_cos_add_sin_mul_I] using congr_arg im (abs_mul_cos_add_sin_mul_I x) theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by refine' ⟨fun hz => ⟨arg z, _⟩, _⟩ · calc exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul] _ = z := abs_mul_exp_arg_mul_I z · rintro ⟨θ, rfl⟩ exact Complex.abs_exp_ofReal_mul_I θ #align complex.abs_eq_one_iff Complex.abs_eq_one_iff @[simp] theorem range_exp_mul_I : (Set.range fun x : ℝ => exp (x * I)) = Metric.sphere 0 1 := by ext x simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, Set.mem_range] set_option linter.uppercaseLean3 false in #align complex.range_exp_mul_I Complex.range_exp_mul_I theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (r * (cos θ + sin θ * I)) = θ := by simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one] simp only [ofReal_mul_re, ofReal_mul_im, neg_im, ← ofReal_cos, ← ofReal_sin, ← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr] by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2) · rw [if_pos] exacts [Real.arcsin_sin' h₁, Real.cos_nonneg_of_mem_Icc h₁] · rw [Set.mem_Icc, not_and_or, not_le, not_le] at h₁ cases' h₁ with h₁ h₁ · replace hθ := hθ.1 have hcos : Real.cos θ < 0 := by rw [← neg_pos, ← Real.cos_add_pi] refine' Real.cos_pos_of_mem_Ioo ⟨_, _⟩ <;> linarith have hsin : Real.sin θ < 0 := Real.sin_neg_of_neg_of_neg_pi_lt (by linarith) hθ rw [if_neg, if_neg, ← Real.sin_add_pi, Real.arcsin_sin, add_sub_cancel] <;> [linarith; linarith; exact hsin.not_le; exact hcos.not_le] · replace hθ := hθ.2 have hcos : Real.cos θ < 0 := Real.cos_neg_of_pi_div_two_lt_of_lt h₁ (by linarith) have hsin : 0 ≤ Real.sin θ := Real.sin_nonneg_of_mem_Icc ⟨by linarith, hθ⟩ rw [if_neg, if_pos, ← Real.sin_sub_pi, Real.arcsin_sin, sub_add_cancel] <;> [linarith; linarith; exact hsin; exact hcos.not_le] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I Complex.arg_mul_cos_add_sin_mul_I theorem arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) : arg (cos θ + sin θ * I) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I Complex.arg_cos_add_sin_mul_I @[simp] theorem arg_zero : arg 0 = 0 := by simp [arg, le_refl] #align complex.arg_zero Complex.arg_zero theorem ext_abs_arg {x y : ℂ} (h₁ : abs x = abs y) (h₂ : x.arg = y.arg) : x = y := by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] #align complex.ext_abs_arg Complex.ext_abs_arg theorem ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y := ⟨fun h => h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ #align complex.ext_abs_arg_iff Complex.ext_abs_arg_iff theorem arg_mem_Ioc (z : ℂ) : arg z ∈ Set.Ioc (-π) π := by have hπ : 0 < π := Real.pi_pos rcases eq_or_ne z 0 with (rfl | hz); simp [hπ, hπ.le] rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩ rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N, ← sin_add_int_mul_two_pi _ N] have := arg_mul_cos_add_sin_mul_I (abs.pos hz) hN push_cast at this rwa [this] #align complex.arg_mem_Ioc Complex.arg_mem_Ioc @[simp] theorem range_arg : Set.range arg = Set.Ioc (-π) π := (Set.range_subset_iff.2 arg_mem_Ioc).antisymm fun _ hx => ⟨_, arg_cos_add_sin_mul_I hx⟩ #align complex.range_arg Complex.range_arg theorem arg_le_pi (x : ℂ) : arg x ≤ π := (arg_mem_Ioc x).2 #align complex.arg_le_pi Complex.arg_le_pi theorem neg_pi_lt_arg (x : ℂ) : -π < arg x := (arg_mem_Ioc x).1 #align complex.neg_pi_lt_arg Complex.neg_pi_lt_arg theorem abs_arg_le_pi (z : ℂ) : |arg z| ≤ π := abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ #align complex.abs_arg_le_pi Complex.abs_arg_le_pi @[simp] theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by rcases eq_or_ne z 0 with (rfl | h₀); · simp calc 0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) := ⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by contrapose! intro h exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩ _ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul] #align complex.arg_nonneg_iff Complex.arg_nonneg_iff @[simp] theorem arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 := lt_iff_lt_of_le_iff_le arg_nonneg_iff #align complex.arg_neg_iff Complex.arg_neg_iff theorem arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x := by rcases eq_or_ne x 0 with (rfl | hx); · rw [mul_zero] conv_lhs => rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← ofReal_mul, arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] #align complex.arg_real_mul Complex.arg_real_mul theorem arg_mul_real {r : ℝ} (hr : 0 < r) (x : ℂ) : arg (x * r) = arg x := mul_comm x r ▸ arg_real_mul x hr theorem arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : arg x = arg y ↔ (abs y / abs x : ℂ) * x = y := by simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_ofReal, abs_abs, div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and_iff] rw [← ofReal_div, arg_real_mul] exact div_pos (abs.pos hy) (abs.pos hx) #align complex.arg_eq_arg_iff Complex.arg_eq_arg_iff @[simp] theorem arg_one : arg 1 = 0 := by simp [arg, zero_le_one] #align complex.arg_one Complex.arg_one @[simp] theorem arg_neg_one : arg (-1) = π := by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] #align complex.arg_neg_one Complex.arg_neg_one @[simp] theorem arg_I : arg I = π / 2 := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_I Complex.arg_I @[simp] theorem arg_neg_I : arg (-I) = -(π / 2) := by simp [arg, le_refl] set_option linter.uppercaseLean3 false in #align complex.arg_neg_I Complex.arg_neg_I @[simp] theorem tan_arg (x : ℂ) : Real.tan (arg x) = x.im / x.re := by by_cases h : x = 0 · simp only [h, zero_div, Complex.zero_im, Complex.arg_zero, Real.tan_zero, Complex.zero_re] rw [Real.tan_eq_sin_div_cos, sin_arg, cos_arg h, div_div_div_cancel_right _ (abs.ne_zero h)] #align complex.tan_arg Complex.tan_arg theorem arg_ofReal_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 := by simp [arg, hx] #align complex.arg_of_real_of_nonneg Complex.arg_ofReal_of_nonneg theorem arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 := by refine' ⟨fun h => _, _⟩ · rw [← abs_mul_cos_add_sin_mul_I z, h] simp [abs.nonneg] · cases' z with x y rintro ⟨h, rfl : y = 0⟩ exact arg_ofReal_of_nonneg h #align complex.arg_eq_zero_iff Complex.arg_eq_zero_iff theorem arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 := by by_cases h₀ : z = 0; simp [h₀, lt_irrefl, Real.pi_ne_zero.symm] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨h : x < 0, rfl : y = 0⟩ rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)] simp [← ofReal_def] #align complex.arg_eq_pi_iff Complex.arg_eq_pi_iff theorem arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 := by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or, not_le, Classical.not_not, arg_eq_pi_iff] #align complex.arg_lt_pi_iff Complex.arg_lt_pi_iff theorem arg_ofReal_of_neg {x : ℝ} (hx : x < 0) : arg x = π := arg_eq_pi_iff.2 ⟨hx, rfl⟩ #align complex.arg_of_real_of_neg Complex.arg_ofReal_of_neg theorem arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : 0 < y⟩ rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, mul_zero, mul_one] #align complex.arg_eq_pi_div_two_iff Complex.arg_eq_pi_div_two_iff theorem arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = -(π / 2) ↔ z.re = 0 ∧ z.im < 0 := by by_cases h₀ : z = 0; · simp [h₀, lt_irrefl, Real.pi_ne_zero] constructor · intro h rw [← abs_mul_cos_add_sin_mul_I z, h] simp [h₀] · cases' z with x y rintro ⟨rfl : x = 0, hy : y < 0⟩ rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I] simp #align complex.arg_eq_neg_pi_div_two_iff Complex.arg_eq_neg_pi_div_two_iff theorem arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = Real.arcsin (x.im / abs x) := if_pos hx #align complex.arg_of_re_nonneg Complex.arg_of_re_nonneg theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) : arg x = Real.arcsin ((-x).im / abs x) + π := by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] #align complex.arg_of_re_neg_of_im_nonneg Complex.arg_of_re_neg_of_im_nonneg theorem arg_of_re_neg_of_im_neg {x : ℂ} (hx_re : x.re < 0) (hx_im : x.im < 0) : arg x = Real.arcsin ((-x).im / abs x) - π := by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] #align complex.arg_of_re_neg_of_im_neg Complex.arg_of_re_neg_of_im_neg theorem arg_of_im_nonneg_of_ne_zero {z : ℂ} (h₁ : 0 ≤ z.im) (h₂ : z ≠ 0) : arg z = Real.arccos (z.re / abs z) := by rw [← cos_arg h₂, Real.arccos_cos (arg_nonneg_iff.2 h₁) (arg_le_pi _)] #align complex.arg_of_im_nonneg_of_ne_zero Complex.arg_of_im_nonneg_of_ne_zero theorem arg_of_im_pos {z : ℂ} (hz : 0 < z.im) : arg z = Real.arccos (z.re / abs z) := arg_of_im_nonneg_of_ne_zero hz.le fun h => hz.ne' <| h.symm ▸ rfl #align complex.arg_of_im_pos Complex.arg_of_im_pos theorem arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -Real.arccos (z.re / abs z) := by have h₀ : z ≠ 0 := mt (congr_arg im) hz.ne rw [← cos_arg h₀, ← Real.cos_neg, Real.arccos_cos, neg_neg] exacts [neg_nonneg.2 (arg_neg_iff.2 hz).le, neg_le.2 (neg_pi_lt_arg z).le] #align complex.arg_of_im_neg Complex.arg_of_im_neg theorem arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x := by simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg, Real.arcsin_neg] rcases lt_trichotomy x.re 0 with (hr | hr | hr) <;> rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hr, hr.not_le, hi.le, hi.ne, not_le.2 hi, add_comm] · simp [hr, hr.not_le, hi] · simp [hr, hr.not_le, hi.ne.symm, hi.le, not_le.2 hi, sub_eq_neg_add] · simp [hr] · simp [hr] · simp [hr] · simp [hr, hr.le, hi.ne] · simp [hr, hr.le, hr.le.not_lt] · simp [hr, hr.le, hr.le.not_lt] #align complex.arg_conj Complex.arg_conj theorem arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x := by rw [← arg_conj, inv_def, mul_comm] by_cases hx : x = 0 · simp [hx] · exact arg_real_mul (conj x) (by simp [hx]) #align complex.arg_inv Complex.arg_inv theorem arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.arcsin_le_pi_div_two, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him.not_lt] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_sub, Real.neg_pi_div_two_lt_arcsin, neg_im, neg_div, neg_lt_neg_iff, div_lt_one, ← _root_.abs_of_nonneg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_neg hre him] exact (sub_le_self _ Real.pi_pos.le).trans (Real.arcsin_le_pi_div_two _) #align complex.arg_le_pi_div_two_iff Complex.arg_le_pi_div_two_iff theorem neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z := by rcases le_or_lt 0 (re z) with hre | hre · simp only [hre, arg_of_re_nonneg hre, Real.neg_pi_div_two_le_arcsin, true_or_iff] simp only [hre.not_le, false_or_iff] rcases le_or_lt 0 (im z) with him | him · simp only [him] rw [iff_true_iff, arg_of_re_neg_of_im_nonneg hre him] exact (Real.neg_pi_div_two_le_arcsin _).trans (le_add_of_nonneg_right Real.pi_pos.le) · simp only [him.not_le] rw [iff_false_iff, not_le, arg_of_re_neg_of_im_neg hre him, sub_lt_iff_lt_add', ← sub_eq_add_neg, sub_half, Real.arcsin_lt_pi_div_two, div_lt_one, neg_im, ← abs_of_neg him, abs_im_lt_abs] exacts [hre.ne, abs.pos <| ne_of_apply_ne re hre.ne] #align complex.neg_pi_div_two_le_arg_iff Complex.neg_pi_div_two_le_arg_iff lemma neg_pi_div_two_lt_arg_iff {z : ℂ} : -(π / 2) < arg z ↔ 0 < re z ∨ 0 ≤ im z := by rw [lt_iff_le_and_ne, neg_pi_div_two_le_arg_iff, ne_comm, Ne, arg_eq_neg_pi_div_two_iff] rcases lt_trichotomy z.re 0 with hre | hre | hre · simp [hre.ne, hre.not_le, hre.not_lt] · simp [hre] · simp [hre, hre.le, hre.ne'] @[simp] theorem abs_arg_le_pi_div_two_iff {z : ℂ} : |arg z| ≤ π / 2 ↔ 0 ≤ re z := by rw [abs_le, arg_le_pi_div_two_iff, neg_pi_div_two_le_arg_iff, ← or_and_left, ← not_le, and_not_self_iff, or_false_iff] #align complex.abs_arg_le_pi_div_two_iff Complex.abs_arg_le_pi_div_two_iff @[simp] theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_conj, h] #align complex.arg_conj_coe_angle Complex.arg_conj_coe_angle @[simp] theorem arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : Real.Angle) = -arg x := by by_cases h : arg x = π <;> simp [arg_inv, h] #align complex.arg_inv_coe_angle Complex.arg_inv_coe_angle theorem arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π := by rw [arg_of_im_pos hi, arg_of_im_neg (show (-x).im < 0 from Left.neg_neg_iff.2 hi)] simp [neg_div, Real.arccos_neg] #align complex.arg_neg_eq_arg_sub_pi_of_im_pos Complex.arg_neg_eq_arg_sub_pi_of_im_pos theorem arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π := by rw [arg_of_im_neg hi, arg_of_im_pos (show 0 < (-x).im from Left.neg_pos_iff.2 hi)] simp [neg_div, Real.arccos_neg, add_comm, ← sub_eq_add_neg] #align complex.arg_neg_eq_arg_add_pi_of_im_neg Complex.arg_neg_eq_arg_add_pi_of_im_neg theorem arg_neg_eq_arg_sub_pi_iff {x : ℂ} : arg (-x) = arg x - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0 := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, hi.ne, hi.not_lt, arg_neg_eq_arg_add_pi_of_im_neg, sub_eq_add_neg, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr] · simp [hr, hi, Real.pi_ne_zero] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr.not_lt, ← add_eq_zero_iff_eq_neg, Real.pi_ne_zero] · simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] #align complex.arg_neg_eq_arg_sub_pi_iff Complex.arg_neg_eq_arg_sub_pi_iff theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} : arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · simp [hi, arg_neg_eq_arg_add_pi_of_im_neg] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le] simp [hr.not_lt, ← two_mul, Real.pi_ne_zero] · simp [hr, hi, Real.pi_ne_zero.symm] · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr)] simp [hr] · simp [hi, hi.ne.symm, hi.not_lt, arg_neg_eq_arg_sub_pi_of_im_pos, sub_eq_add_neg, ← add_eq_zero_iff_neg_eq, Real.pi_ne_zero] #align complex.arg_neg_eq_arg_add_pi_iff Complex.arg_neg_eq_arg_add_pi_iff theorem arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : Real.Angle) = arg x + π := by rcases lt_trichotomy x.im 0 with (hi | hi | hi) · rw [arg_neg_eq_arg_add_pi_of_im_neg hi, Real.Angle.coe_add] · rw [(ext rfl hi : x = x.re)] rcases lt_trichotomy x.re 0 with (hr | hr | hr) · rw [arg_ofReal_of_neg hr, ← ofReal_neg, arg_ofReal_of_nonneg (Left.neg_pos_iff.2 hr).le, ← Real.Angle.coe_add, ← two_mul, Real.Angle.coe_two_pi, Real.Angle.coe_zero] · exact False.elim (hx (ext hr hi)) · rw [arg_ofReal_of_nonneg hr.le, ← ofReal_neg, arg_ofReal_of_neg (Left.neg_neg_iff.2 hr), Real.Angle.coe_zero, zero_add] · rw [arg_neg_eq_arg_sub_pi_of_im_pos hi, Real.Angle.coe_sub, Real.Angle.sub_coe_pi_eq_add_coe_pi] #align complex.arg_neg_coe_angle Complex.arg_neg_coe_angle theorem arg_mul_cos_add_sin_mul_I_eq_toIocMod {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) = toIocMod Real.two_pi_pos (-π) θ := by have hi : toIocMod Real.two_pi_pos (-π) θ ∈ Set.Ioc (-π) π := by convert toIocMod_mem_Ioc _ _ θ ring convert arg_mul_cos_add_sin_mul_I hr hi using 3 simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi] set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_mul_cos_add_sin_mul_I_eq_toIocMod theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) : arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod Complex.arg_cos_add_sin_mul_I_eq_toIocMod theorem arg_mul_cos_add_sin_mul_I_sub {r : ℝ} (hr : 0 < r) (θ : ℝ) : arg (r * (cos θ + sin θ * I)) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [arg_mul_cos_add_sin_mul_I_eq_toIocMod hr, toIocMod_sub_self, toIocDiv_eq_neg_floor, zsmul_eq_mul] ring_nf set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_sub Complex.arg_mul_cos_add_sin_mul_I_sub theorem arg_cos_add_sin_mul_I_sub (θ : ℝ) : arg (cos θ + sin θ * I) - θ = 2 * π * ⌊(π - θ) / (2 * π)⌋ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_sub Complex.arg_cos_add_sin_mul_I_sub theorem arg_mul_cos_add_sin_mul_I_coe_angle {r : ℝ} (hr : 0 < r) (θ : Real.Angle) : (arg (r * (Real.Angle.cos θ + Real.Angle.sin θ * I)) : Real.Angle) = θ := by induction' θ using Real.Angle.induction_on with θ rw [Real.Angle.cos_coe, Real.Angle.sin_coe, Real.Angle.angle_eq_iff_two_pi_dvd_sub] use ⌊(π - θ) / (2 * π)⌋ exact mod_cast arg_mul_cos_add_sin_mul_I_sub hr θ set_option linter.uppercaseLean3 false in #align complex.arg_mul_cos_add_sin_mul_I_coe_angle Complex.arg_mul_cos_add_sin_mul_I_coe_angle theorem arg_cos_add_sin_mul_I_coe_angle (θ : Real.Angle) : (arg (Real.Angle.cos θ + Real.Angle.sin θ * I) : Real.Angle) = θ := by rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] set_option linter.uppercaseLean3 false in #align complex.arg_cos_add_sin_mul_I_coe_angle Complex.arg_cos_add_sin_mul_I_coe_angle theorem arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x * y) : Real.Angle) = arg x + arg y := by convert arg_mul_cos_add_sin_mul_I_coe_angle (mul_pos (abs.pos hx) (abs.pos hy)) (arg x + arg y : Real.Angle) using 3 simp_rw [← Real.Angle.coe_add, Real.Angle.sin_coe, Real.Angle.cos_coe, ofReal_cos, ofReal_sin, cos_add_sin_I, ofReal_add, add_mul, exp_add, ofReal_mul] rw [mul_assoc, mul_comm (exp _), ← mul_assoc (abs y : ℂ), abs_mul_exp_arg_mul_I, mul_comm y, ← mul_assoc, abs_mul_exp_arg_mul_I] #align complex.arg_mul_coe_angle Complex.arg_mul_coe_angle theorem arg_div_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) : (arg (x / y) : Real.Angle) = arg x - arg y := by rw [div_eq_mul_inv, arg_mul_coe_angle hx (inv_ne_zero hy), arg_inv_coe_angle, sub_eq_add_neg] #align complex.arg_div_coe_angle Complex.arg_div_coe_angle @[simp] theorem arg_coe_angle_toReal_eq_arg (z : ℂ) : (arg z : Real.Angle).toReal = arg z := by rw [Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] exact arg_mem_Ioc _ #align complex.arg_coe_angle_to_real_eq_arg Complex.arg_coe_angle_toReal_eq_arg theorem arg_coe_angle_eq_iff_eq_toReal {z : ℂ} {θ : Real.Angle} : (arg z : Real.Angle) = θ ↔ arg z = θ.toReal := by rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff_eq_to_real Complex.arg_coe_angle_eq_iff_eq_toReal @[simp] theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg] #align complex.arg_coe_angle_eq_iff Complex.arg_coe_angle_eq_iff lemma arg_mul_eq_add_arg_iff {x y : ℂ} (hx₀ : x ≠ 0) (hy₀ : y ≠ 0) : (x * y).arg = x.arg + y.arg ↔ arg x + arg y ∈ Set.Ioc (-π) π := by rw [← arg_coe_angle_toReal_eq_arg, arg_mul_coe_angle hx₀ hy₀, ← Real.Angle.coe_add, Real.Angle.toReal_coe_eq_self_iff_mem_Ioc] alias ⟨_, arg_mul⟩ := arg_mul_eq_add_arg_iff section Continuity variable {x z : ℂ} theorem arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] fun x => Real.arcsin (x.im / abs x) := ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono fun _ hy => arg_of_re_nonneg hy.le #align complex.arg_eq_nhds_of_re_pos Complex.arg_eq_nhds_of_re_pos theorem arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) + π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_im) #align complex.arg_eq_nhds_of_re_neg_of_im_pos Complex.arg_eq_nhds_of_re_neg_of_im_pos theorem arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) : arg =ᶠ[𝓝 x] fun x => Real.arcsin ((-x).im / abs x) - π := by suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ y.im < 0 exact h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_neg hy.1 hy.2 refine' IsOpen.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0) exact IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_im continuous_zero) #align complex.arg_eq_nhds_of_re_neg_of_im_neg Complex.arg_eq_nhds_of_re_neg_of_im_neg theorem arg_eq_nhds_of_im_pos (hz : 0 < im z) : arg =ᶠ[𝓝 z] fun x => Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono fun _ => arg_of_im_pos #align complex.arg_eq_nhds_of_im_pos Complex.arg_eq_nhds_of_im_pos theorem arg_eq_nhds_of_im_neg (hz : im z < 0) : arg =ᶠ[𝓝 z] fun x => -Real.arccos (x.re / abs x) := ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono fun _ => arg_of_im_neg #align complex.arg_eq_nhds_of_im_neg Complex.arg_eq_nhds_of_im_neg theorem continuousAt_arg (h : 0 < x.re ∨ x.im ≠ 0) : ContinuousAt arg x := by have h₀ : abs x ≠ 0 := by rw [abs.ne_zero_iff] rintro rfl simp at h rw [← lt_or_lt_iff_ne] at h rcases h with (hx_re | hx_im | hx_im) exacts [(Real.continuousAt_arcsin.comp (continuous_im.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).neg.congr (arg_eq_nhds_of_im_neg hx_im).symm, (Real.continuous_arccos.continuousAt.comp (continuous_re.continuousAt.div continuous_abs.continuousAt h₀)).congr (arg_eq_nhds_of_im_pos hx_im).symm] #align complex.continuous_at_arg Complex.continuousAt_arg theorem tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : Tendsto arg (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) := by suffices H : Tendsto (fun x : ℂ => Real.arcsin ((-x).im / abs x) - π) (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 (-π)) · refine' H.congr' _ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) -- Porting note: need to specify the `nhdsWithin` set filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | z.im < 0 }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_neg him.not_le] convert (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div -- Porting note: added type hint to assist in goal state below continuous_abs.continuousWithinAt (s := { z : ℂ | z.im < 0 }) (_ : abs z ≠ 0)) -- Porting note: specify constant precisely to assist in goal below ).sub_const π using 1 · simp [him] · lift z to ℝ using him simpa using hre.ne #align complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero Complex.tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_pos him] refine' ContinuousWithinAt.congr_of_eventuallyEq _ this _ ·

refine' (Real.continuousAt_arcsin.comp_continuousWithinAt ((continuous_im.continuousAt.comp_continuousWithinAt continuousWithinAt_neg).div continuous_abs.continuousWithinAt _)).add tendsto_const_nhds

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z := by have : arg =ᶠ[𝓝[{ z : ℂ | 0 ≤ z.im }] z] fun x => Real.arcsin ((-x).im / abs x) + π := by have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0 := continuous_re.tendsto z (gt_mem_nhds hre) filter_upwards [self_mem_nhdsWithin (s := { z : ℂ | 0 ≤ z.im }), mem_nhdsWithin_of_mem_nhds this] with _ him hre rw [arg, if_neg hre.not_le, if_pos him] refine' ContinuousWithinAt.congr_of_eventuallyEq _ this _ ·

Mathlib.Analysis.SpecialFunctions.Complex.Arg.597_0.CflASCTDE9UCom5

theorem continuousWithinAt_arg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) : ContinuousWithinAt arg { z : ℂ | 0 ≤ z.im } z

Mathlib_Analysis_SpecialFunctions_Complex_Arg