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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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arg_eq_neg_pi_div_two_iff {z : ℂ} : arg z = - (π / 2) ↔ z.re = 0 ∧ z.im < 0 | begin
by_cases h₀ : z = 0, { simp [h₀, lt_irrefl, real.pi_ne_zero] },
split,
{ 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 }
end | lemma | complex.arg_eq_neg_pi_div_two_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.pi_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_of_re_nonneg {x : ℂ} (hx : 0 ≤ x.re) : arg x = real.arcsin (x.im / x.abs) | if_pos hx | lemma | complex.arg_of_re_nonneg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 / x.abs) + π | by simp only [arg, hx_re.not_le, hx_im, if_true, if_false] | lemma | complex.arg_of_re_neg_of_im_nonneg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 / x.abs) - π | by simp only [arg, hx_re.not_le, hx_im.not_le, if_false] | lemma | complex.arg_of_re_neg_of_im_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 _)] | lemma | complex.arg_of_im_nonneg_of_ne_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arccos",
"real.arccos_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 (λ h, hz.ne' $ h.symm ▸ rfl) | lemma | complex.arg_of_im_pos | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arccos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_of_im_neg {z : ℂ} (hz : z.im < 0) : arg z = -real.arccos (z.re / abs z) | begin
have h₀ : z ≠ 0, from 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]
end | lemma | complex.arg_of_im_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arccos",
"real.arccos_cos",
"real.cos_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_conj (x : ℂ) : arg (conj x) = if arg x = π then π else -arg x | begin
simp_rw [arg_eq_pi_iff, arg, neg_im, conj_im, conj_re, abs_conj, neg_div, neg_neg,
real.arcsin_neg, apply_ite has_neg.neg, neg_add, neg_sub, neg_neg, ←sub_eq_add_neg,
sub_neg_eq_add, add_comm π],
rcases lt_trichotomy x.re 0 with (hr|hr|hr); rcases lt_trichotomy x.im 0 with (hi|hi|hi),
... | lemma | complex.arg_conj | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"apply_ite",
"neg_div",
"real.arcsin_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_inv (x : ℂ) : arg x⁻¹ = if arg x = π then π else -arg x | begin
rw [←arg_conj, inv_def, mul_comm],
by_cases hx : x = 0,
{ simp [hx] },
{ exact arg_real_mul (conj x) (by simp [hx]) }
end | lemma | complex.arg_inv | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_le_pi_div_two_iff {z : ℂ} : arg z ≤ π / 2 ↔ 0 ≤ re z ∨ im z < 0 | begin
cases 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] },
simp only [hre.not_le, false_or],
cases le_or_lt 0 (im z) with him him,
{ simp only [him.not_lt],
rw [iff_false, not_le, arg_of_re_neg_of_im_nonneg hre him, ← sub_lt_iff_lt_add, half_... | lemma | complex.arg_le_pi_div_two_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"div_lt_one",
"half_sub",
"ne_of_apply_ne",
"neg_div",
"real.arcsin_le_pi_div_two",
"real.neg_pi_div_two_lt_arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_div_two_le_arg_iff {z : ℂ} : -(π / 2) ≤ arg z ↔ 0 ≤ re z ∨ 0 ≤ im z | begin
cases 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] },
simp only [hre.not_le, false_or],
cases le_or_lt 0 (im z) with him him,
{ simp only [him],
rw [iff_true, arg_of_re_neg_of_im_nonneg hre him],
exact (real.neg_pi_div_two_le_arc... | lemma | complex.neg_pi_div_two_le_arg_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"abs_of_neg",
"div_lt_one",
"ne_of_apply_ne",
"real.arcsin_lt_pi_div_two",
"real.neg_pi_div_two_le_arcsin",
"sub_half"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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_distrib_left, ← not_le,
and_not_self, or_false] | lemma | complex.abs_arg_le_pi_div_two_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"abs_le",
"or_and_distrib_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_conj_coe_angle (x : ℂ) : (arg (conj x) : real.angle) = -arg x | begin
by_cases h : arg x = π;
simp [arg_conj, h]
end | lemma | complex.arg_conj_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_inv_coe_angle (x : ℂ) : (arg x⁻¹ : real.angle) = -arg x | begin
by_cases h : arg x = π;
simp [arg_inv, h]
end | lemma | complex.arg_inv_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_eq_arg_sub_pi_of_im_pos {x : ℂ} (hi : 0 < x.im) : arg (-x) = arg x - π | begin
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]
end | lemma | complex.arg_neg_eq_arg_sub_pi_of_im_pos | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"neg_div",
"real.arccos_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_eq_arg_add_pi_of_im_neg {x : ℂ} (hi : x.im < 0) : arg (-x) = arg x + π | begin
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]
end | lemma | complex.arg_neg_eq_arg_add_pi_of_im_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"neg_div",
"real.arccos_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_eq_arg_sub_pi_iff {x : ℂ} :
arg (-x) = arg x - π ↔ (0 < x.im ∨ x.im = 0 ∧ x.re < 0) | begin
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_of_real_of_neg hr, ←of_real_neg, arg... | lemma | complex.arg_neg_eq_arg_sub_pi_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.pi_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_eq_arg_add_pi_iff {x : ℂ} :
arg (-x) = arg x + π ↔ (x.im < 0 ∨ x.im = 0 ∧ 0 < x.re) | begin
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_of_real_of_neg hr, ←of_real_neg, arg_of_real_of_nonneg (left.neg_pos_iff.2 hr).le],
simp [hr.not_lt, ←two_mul, real.... | lemma | complex.arg_neg_eq_arg_add_pi_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.pi_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_coe_angle {x : ℂ} (hx : x ≠ 0) : (arg (-x) : real.angle) = arg x + π | begin
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_of_real_of_neg hr, ←of_real_neg, arg_of_real_of_nonneg (left.neg_pos_iff.2 hr).le,
←real.angle.... | lemma | complex.arg_neg_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle",
"real.angle.coe_add",
"real.angle.coe_sub",
"real.angle.coe_two_pi",
"real.angle.coe_zero",
"real.angle.sub_coe_pi_eq_add_coe_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod {r : ℝ} (hr : 0 < r) (θ : ℝ) :
arg (r * (cos θ + sin θ * I)) = to_Ioc_mod real.two_pi_pos (-π) θ | begin
have hi : to_Ioc_mod real.two_pi_pos (-π) θ ∈ Ioc (-π) π,
{ convert to_Ioc_mod_mem_Ioc _ _ _,
ring },
convert arg_mul_cos_add_sin_mul_I hr hi using 3,
simp [to_Ioc_mod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi]
end | lemma | complex.arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.two_pi_pos",
"ring",
"to_Ioc_mod",
"to_Ioc_mod_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_cos_add_sin_mul_I_eq_to_Ioc_mod (θ : ℝ) :
arg (cos θ + sin θ * I) = to_Ioc_mod real.two_pi_pos (-π) θ | by rw [←one_mul (_ + _), ←of_real_one, arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod zero_lt_one] | lemma | complex.arg_cos_add_sin_mul_I_eq_to_Ioc_mod | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.two_pi_pos",
"to_Ioc_mod",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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)⌋ | begin
rw [arg_mul_cos_add_sin_mul_I_eq_to_Ioc_mod hr, to_Ioc_mod_sub_self, to_Ioc_div_eq_neg_floor,
zsmul_eq_mul],
ring_nf
end | lemma | complex.arg_mul_cos_add_sin_mul_I_sub | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"to_Ioc_div_eq_neg_floor",
"to_Ioc_mod_sub_self",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 (_ + _), ←of_real_one, arg_mul_cos_add_sin_mul_I_sub zero_lt_one] | lemma | complex.arg_cos_add_sin_mul_I_sub | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 | begin
induction θ using real.angle.induction_on,
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
end | lemma | complex.arg_mul_cos_add_sin_mul_I_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle",
"real.angle.angle_eq_iff_two_pi_dvd_sub",
"real.angle.cos",
"real.angle.cos_coe",
"real.angle.induction_on",
"real.angle.sin",
"real.angle.sin_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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 (_ + _), ←of_real_one, arg_mul_cos_add_sin_mul_I_coe_angle zero_lt_one] | lemma | complex.arg_cos_add_sin_mul_I_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle",
"real.angle.cos",
"real.angle.sin",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_mul_coe_angle {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
(arg (x * y) : real.angle) = arg x + arg y | begin
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, of_real_cos,
of_real_sin, cos_add_sin_I, of_real_add, add_mul, ex... | lemma | complex.arg_mul_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"exp",
"exp_add",
"mul_assoc",
"mul_comm",
"real.angle",
"real.angle.cos_coe",
"real.angle.sin_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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] | lemma | complex.arg_div_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"div_eq_mul_inv",
"inv_ne_zero",
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_coe_angle_to_real_eq_arg (z : ℂ) : (arg z : real.angle).to_real = arg z | begin
rw real.angle.to_real_coe_eq_self_iff_mem_Ioc,
exact arg_mem_Ioc _
end | lemma | complex.arg_coe_angle_to_real_eq_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle",
"real.angle.to_real_coe_eq_self_iff_mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_coe_angle_eq_iff_eq_to_real {z : ℂ} {θ : real.angle} :
(arg z : real.angle) = θ ↔ arg z = θ.to_real | by rw [←real.angle.to_real_inj, arg_coe_angle_to_real_eq_arg] | lemma | complex.arg_coe_angle_eq_iff_eq_to_real | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_coe_angle_eq_iff {x y : ℂ} : (arg x : real.angle) = arg y ↔ arg x = arg y | by simp_rw [←real.angle.to_real_inj, arg_coe_angle_to_real_eq_arg] | lemma | complex.arg_coe_angle_eq_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_nhds_of_re_pos (hx : 0 < x.re) : arg =ᶠ[𝓝 x] λ x, real.arcsin (x.im / x.abs) | ((continuous_re.tendsto _).eventually (lt_mem_nhds hx)).mono $ λ y hy, arg_of_re_nonneg hy.le | lemma | complex.arg_eq_nhds_of_re_pos | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"lt_mem_nhds",
"real.arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_nhds_of_re_neg_of_im_pos (hx_re : x.re < 0) (hx_im : 0 < x.im) :
arg =ᶠ[𝓝 x] λ x, real.arcsin ((-x).im / x.abs) + π | begin
suffices h_forall_nhds : ∀ᶠ (y : ℂ) in (𝓝 x), y.re < 0 ∧ 0 < y.im,
from h_forall_nhds.mono (λ y hy, arg_of_re_neg_of_im_nonneg hy.1 hy.2.le),
refine is_open.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im),
exact is_open.and (is_open_lt continuous_re continuous_zero)
(is_open_lt continuous_z... | lemma | complex.arg_eq_nhds_of_re_neg_of_im_pos | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"is_open.and",
"is_open.eventually_mem",
"is_open_lt",
"real.arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_nhds_of_re_neg_of_im_neg (hx_re : x.re < 0) (hx_im : x.im < 0) :
arg =ᶠ[𝓝 x] λ x, real.arcsin ((-x).im / x.abs) - π | begin
suffices h_forall_nhds : ∀ᶠ (y : ℂ) in (𝓝 x), y.re < 0 ∧ y.im < 0,
from h_forall_nhds.mono (λ y hy, arg_of_re_neg_of_im_neg hy.1 hy.2),
refine is_open.eventually_mem _ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ x.im < 0),
exact is_open.and (is_open_lt continuous_re continuous_zero)
(is_open_lt continuous_im cont... | lemma | complex.arg_eq_nhds_of_re_neg_of_im_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"is_open.and",
"is_open.eventually_mem",
"is_open_lt",
"real.arcsin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_nhds_of_im_pos (hz : 0 < im z) :
arg =ᶠ[𝓝 z] λ x, real.arccos (x.re / abs x) | ((continuous_im.tendsto _).eventually (lt_mem_nhds hz)).mono $ λ x, arg_of_im_pos | lemma | complex.arg_eq_nhds_of_im_pos | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"lt_mem_nhds",
"real.arccos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_nhds_of_im_neg (hz : im z < 0) :
arg =ᶠ[𝓝 z] λ x, -real.arccos (x.re / abs x) | ((continuous_im.tendsto _).eventually (gt_mem_nhds hz)).mono $ λ x, arg_of_im_neg | lemma | complex.arg_eq_nhds_of_im_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"gt_mem_nhds",
"real.arccos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_arg (h : 0 < x.re ∨ x.im ≠ 0) : continuous_at arg x | begin
have h₀ : abs x ≠ 0, { rw abs.ne_zero_iff, rintro rfl, simpa using h },
rw [← lt_or_lt_iff_ne] at h,
rcases h with (hx_re|hx_im|hx_im),
exacts [(real.continuous_at_arcsin.comp (continuous_im.continuous_at.div
continuous_abs.continuous_at h₀)).congr (arg_eq_nhds_of_re_pos hx_re).symm,
(real.continu... | lemma | complex.continuous_at_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"continuous_at",
"lt_or_lt_iff_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_arg_nhds_within_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)) | begin
suffices H :
tendsto (λ x : ℂ, real.arcsin ((-x).im / x.abs) - π) (𝓝[{z : ℂ | z.im < 0}] z) (𝓝 (-π)),
{ refine H.congr' _,
have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0, from continuous_re.tendsto z (gt_mem_nhds hre),
filter_upwards [self_mem_nhds_within, mem_nhds_within_of_mem_nhds this] with _ him hre,
... | lemma | complex.tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"gt_mem_nhds",
"lift",
"mem_nhds_within_of_mem_nhds",
"real.arcsin",
"self_mem_nhds_within",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_arg_of_re_neg_of_im_zero
{z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
continuous_within_at arg {z : ℂ | 0 ≤ z.im} z | begin
have : arg =ᶠ[𝓝[{z : ℂ | 0 ≤ z.im}] z] λ x, real.arcsin ((-x).im / x.abs) + π,
{ have : ∀ᶠ x : ℂ in 𝓝 z, x.re < 0, from continuous_re.tendsto z (gt_mem_nhds hre),
filter_upwards [self_mem_nhds_within, mem_nhds_within_of_mem_nhds this] with _ him hre,
rw [arg, if_neg hre.not_le, if_pos him] },
refi... | lemma | complex.continuous_within_at_arg_of_re_neg_of_im_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"continuous_within_at",
"continuous_within_at.congr_of_eventually_eq",
"gt_mem_nhds",
"lift",
"mem_nhds_within_of_mem_nhds",
"real.arcsin",
"self_mem_nhds_within",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_arg_nhds_within_im_nonneg_of_re_neg_of_im_zero
{z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
tendsto arg (𝓝[{z : ℂ | 0 ≤ z.im}] z) (𝓝 π) | by simpa only [arg_eq_pi_iff.2 ⟨hre, him⟩]
using (continuous_within_at_arg_of_re_neg_of_im_zero hre him).tendsto | lemma | complex.tendsto_arg_nhds_within_im_nonneg_of_re_neg_of_im_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_arg_coe_angle (h : x ≠ 0) : continuous_at (coe ∘ arg : ℂ → real.angle) x | begin
by_cases hs : 0 < x.re ∨ x.im ≠ 0,
{ exact real.angle.continuous_coe.continuous_at.comp (continuous_at_arg hs) },
{ rw [←function.comp.right_id (coe ∘ arg),
(function.funext_iff.2 (λ _, (neg_neg _).symm) :
(id : ℂ → ℂ) = has_neg.neg ∘ has_neg.neg), ←function.comp.assoc],
refine continu... | lemma | complex.continuous_at_arg_coe_angle | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_at_const",
"continuous_at_update_of_ne",
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective_arg : injective (λ z : circle, arg z) | λ z w h, subtype.ext $ ext_abs_arg ((abs_coe_circle z).trans (abs_coe_circle w).symm) h | lemma | circle.injective_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"abs_coe_circle",
"circle",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_arg {z w : circle} : arg z = arg w ↔ z = w | injective_arg.eq_iff | lemma | circle.arg_eq_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_exp_map_circle {x : ℝ} (h₁ : -π < x) (h₂ : x ≤ π) : arg (exp_map_circle x) = x | by rw [exp_map_circle_apply, exp_mul_I, arg_cos_add_sin_mul_I ⟨h₁, h₂⟩] | lemma | arg_exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"exp_map_circle_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_arg (z : circle) : exp_map_circle (arg z) = z | circle.injective_arg $ arg_exp_map_circle (neg_pi_lt_arg _) (arg_le_pi _) | lemma | exp_map_circle_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"arg_exp_map_circle",
"circle",
"circle.injective_arg",
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_local_equiv : local_equiv circle ℝ | { to_fun := arg ∘ coe,
inv_fun := exp_map_circle,
source := univ,
target := Ioc (-π) π,
map_source' := λ z _, ⟨neg_pi_lt_arg _, arg_le_pi _⟩,
map_target' := maps_to_univ _ _,
left_inv' := λ z _, exp_map_circle_arg z,
right_inv' := λ x hx, arg_exp_map_circle hx.1 hx.2 } | def | circle.arg_local_equiv | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"arg_exp_map_circle",
"circle",
"exp_map_circle",
"exp_map_circle_arg",
"inv_fun",
"local_equiv"
] | `complex.arg ∘ coe` and `exp_map_circle` define a local equivalence between `circle and `ℝ` with
`source = set.univ` and `target = set.Ioc (-π) π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arg_equiv : circle ≃ Ioc (-π) π | { to_fun := λ z, ⟨arg z, neg_pi_lt_arg _, arg_le_pi _⟩,
inv_fun := exp_map_circle ∘ coe,
left_inv := λ z, arg_local_equiv.left_inv trivial,
right_inv := λ x, subtype.ext $ arg_local_equiv.right_inv x.2 } | def | circle.arg_equiv | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"circle",
"exp_map_circle",
"inv_fun",
"subtype.ext"
] | `complex.arg` and `exp_map_circle` define an equivalence between `circle and `(-π, π]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
left_inverse_exp_map_circle_arg : left_inverse exp_map_circle (arg ∘ coe) | exp_map_circle_arg | lemma | left_inverse_exp_map_circle_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"exp_map_circle_arg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_on_arg_exp_map_circle : inv_on (arg ∘ coe) exp_map_circle (Ioc (-π) π) univ | circle.arg_local_equiv.symm.inv_on | lemma | inv_on_arg_exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_exp_map_circle_neg_pi_pi : surj_on exp_map_circle (Ioc (-π) π) univ | circle.arg_local_equiv.symm.surj_on | lemma | surj_on_exp_map_circle_neg_pi_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_eq_exp_map_circle {x y : ℝ} :
exp_map_circle x = exp_map_circle y ↔ ∃ m : ℤ, x = y + m * (2 * π) | begin
rw [subtype.ext_iff, exp_map_circle_apply, exp_map_circle_apply, exp_eq_exp_iff_exists_int],
refine exists_congr (λ n, _),
rw [← mul_assoc, ← add_mul, mul_left_inj' I_ne_zero, ← of_real_one, ← of_real_bit0,
← of_real_mul, ← of_real_int_cast, ← of_real_mul, ← of_real_add, of_real_inj]
end | lemma | exp_map_circle_eq_exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"exp_map_circle_apply",
"mul_assoc",
"mul_left_inj'",
"subtype.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
periodic_exp_map_circle : periodic exp_map_circle (2 * π) | λ z, exp_map_circle_eq_exp_map_circle.2 ⟨1, by rw [int.cast_one, one_mul]⟩ | lemma | periodic_exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"int.cast_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_two_pi : exp_map_circle (2 * π) = 1 | periodic_exp_map_circle.eq.trans exp_map_circle_zero | lemma | exp_map_circle_two_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"exp_map_circle_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_sub_two_pi (x : ℝ) : exp_map_circle (x - 2 * π) = exp_map_circle x | periodic_exp_map_circle.sub_eq x | lemma | exp_map_circle_sub_two_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_map_circle_add_two_pi (x : ℝ) : exp_map_circle (x + 2 * π) = exp_map_circle x | periodic_exp_map_circle x | lemma | exp_map_circle_add_two_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"periodic_exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.angle.exp_map_circle (θ : real.angle) : circle | periodic_exp_map_circle.lift θ | def | real.angle.exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"circle",
"real.angle"
] | `exp_map_circle`, applied to a `real.angle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.angle.exp_map_circle_coe (x : ℝ) :
real.angle.exp_map_circle x = exp_map_circle x | rfl | lemma | real.angle.exp_map_circle_coe | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle",
"real.angle.exp_map_circle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.angle.coe_exp_map_circle (θ : real.angle) : (θ.exp_map_circle : ℂ) = θ.cos + θ.sin * I | begin
induction θ using real.angle.induction_on,
simp [complex.exp_mul_I],
end | lemma | real.angle.coe_exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"complex.exp_mul_I",
"real.angle",
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.angle.exp_map_circle_zero :
real.angle.exp_map_circle 0 = 1 | by rw [←real.angle.coe_zero, real.angle.exp_map_circle_coe, exp_map_circle_zero] | lemma | real.angle.exp_map_circle_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle_zero",
"real.angle.exp_map_circle",
"real.angle.exp_map_circle_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.angle.exp_map_circle_neg (θ : real.angle) :
real.angle.exp_map_circle (-θ) = (real.angle.exp_map_circle θ)⁻¹ | begin
induction θ using real.angle.induction_on,
simp_rw [←real.angle.coe_neg, real.angle.exp_map_circle_coe, exp_map_circle_neg]
end | lemma | real.angle.exp_map_circle_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle_neg",
"real.angle",
"real.angle.exp_map_circle",
"real.angle.exp_map_circle_coe",
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.angle.exp_map_circle_add (θ₁ θ₂ : real.angle) :
real.angle.exp_map_circle (θ₁ + θ₂) =
(real.angle.exp_map_circle θ₁) * (real.angle.exp_map_circle θ₂) | begin
induction θ₁ using real.angle.induction_on,
induction θ₂ using real.angle.induction_on,
exact exp_map_circle_add θ₁ θ₂
end | lemma | real.angle.exp_map_circle_add | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle_add",
"real.angle",
"real.angle.exp_map_circle",
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.angle.arg_exp_map_circle (θ : real.angle) :
(arg (real.angle.exp_map_circle θ) : real.angle) = θ | begin
induction θ using real.angle.induction_on,
rw [real.angle.exp_map_circle_coe, exp_map_circle_apply, exp_mul_I, ←of_real_cos,
←of_real_sin, ←real.angle.cos_coe, ←real.angle.sin_coe, arg_cos_add_sin_mul_I_coe_angle]
end | lemma | real.angle.arg_exp_map_circle | analysis.special_functions.complex | src/analysis/special_functions/complex/circle.lean | [
"analysis.complex.circle",
"analysis.special_functions.complex.log"
] | [
"exp_map_circle_apply",
"real.angle",
"real.angle.exp_map_circle",
"real.angle.exp_map_circle_coe",
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log (x : ℂ) : ℂ | x.abs.log + arg x * I | def | complex.log | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | Inverse of the `exp` function. Returns values such that `(log x).im > - π` and `(log x).im ≤ π`.
`log 0 = 0` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_re (x : ℂ) : x.log.re = x.abs.log | by simp [log] | lemma | complex.log_re | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_im (x : ℂ) : x.log.im = x.arg | by simp [log] | lemma | complex.log_im | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_lt_log_im (x : ℂ) : -π < (log x).im | by simp only [log_im, neg_pi_lt_arg] | lemma | complex.neg_pi_lt_log_im | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_im_le_pi (x : ℂ) : (log x).im ≤ π | by simp only [log_im, arg_le_pi] | lemma | complex.log_im_le_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_log {x : ℂ} (hx : x ≠ 0) : exp (log x) = x | by rw [log, exp_add_mul_I, ← of_real_sin, sin_arg, ← of_real_cos, cos_arg hx,
← of_real_exp, real.exp_log (abs.pos hx), mul_add, of_real_div, of_real_div,
mul_div_cancel' _ (of_real_ne_zero.2 $ abs.ne_zero hx), ← mul_assoc,
mul_div_cancel' _ (of_real_ne_zero.2 $ abs.ne_zero hx), re_add_im] | lemma | complex.exp_log | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp",
"mul_assoc",
"mul_div_cancel'",
"real.exp_log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_exp : range exp = {0}ᶜ | set.ext $ λ x, ⟨by { rintro ⟨x, rfl⟩, exact exp_ne_zero x }, λ hx, ⟨log x, exp_log hx⟩⟩ | lemma | complex.range_exp | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_exp {x : ℂ} (hx₁ : -π < x.im) (hx₂: x.im ≤ π) : log (exp x) = x | by rw [log, abs_exp, real.log_exp, exp_eq_exp_re_mul_sin_add_cos, ← of_real_exp,
arg_mul_cos_add_sin_mul_I (real.exp_pos _) ⟨hx₁, hx₂⟩, re_add_im] | lemma | complex.log_exp | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp",
"real.exp_pos",
"real.log_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_inj_of_neg_pi_lt_of_le_pi {x y : ℂ} (hx₁ : -π < x.im) (hx₂ : x.im ≤ π)
(hy₁ : - π < y.im) (hy₂ : y.im ≤ π) (hxy : exp x = exp y) : x = y | by rw [← log_exp hx₁ hx₂, ← log_exp hy₁ hy₂, hxy] | lemma | complex.exp_inj_of_neg_pi_lt_of_le_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_real_log {x : ℝ} (hx : 0 ≤ x) : (x.log : ℂ) = log x | complex.ext
(by rw [log_re, of_real_re, abs_of_nonneg hx])
(by rw [of_real_im, log_im, arg_of_real_of_nonneg hx]) | lemma | complex.of_real_log | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"abs_of_nonneg",
"complex.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_of_real_re (x : ℝ) : (log (x : ℂ)).re = real.log x | by simp [log_re] | lemma | complex.log_of_real_re | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"real.log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_of_real_mul {r : ℝ} (hr : 0 < r) {x : ℂ} (hx : x ≠ 0) :
log (r * x) = real.log r + log x | begin
replace hx := complex.abs.ne_zero_iff.mpr hx,
simp_rw [log, map_mul, abs_of_real, arg_real_mul _ hr, abs_of_pos hr, real.log_mul hr.ne' hx,
of_real_add, add_assoc],
end | lemma | complex.log_of_real_mul | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"abs_of_pos",
"map_mul",
"real.log",
"real.log_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_mul_of_real (r : ℝ) (hr : 0 < r) (x : ℂ) (hx : x ≠ 0) :
log (x * r) = real.log r + log x | by rw [mul_comm, log_of_real_mul hr hx, add_comm] | lemma | complex.log_mul_of_real | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"mul_comm",
"real.log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_zero : log 0 = 0 | by simp [log] | lemma | complex.log_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_one : log 1 = 0 | by simp [log] | lemma | complex.log_one | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_neg_one : log (-1) = π * I | by simp [log] | lemma | complex.log_neg_one | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_I : log I = π / 2 * I | by simp [log] | lemma | complex.log_I | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_neg_I : log (-I) = -(π / 2) * I | by simp [log] | lemma | complex.log_neg_I | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_conj_eq_ite (x : ℂ) :
log (conj x) = if x.arg = π then log x else conj (log x) | begin
simp_rw [log, abs_conj, arg_conj, map_add, map_mul, conj_of_real],
split_ifs with hx,
{ rw hx },
simp_rw [of_real_neg, conj_I, mul_neg, neg_mul]
end | lemma | complex.log_conj_eq_ite | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"map_mul",
"mul_neg",
"neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_conj (x : ℂ) (h : x.arg ≠ π) : log (conj x) = conj (log x) | by rw [log_conj_eq_ite, if_neg h] | lemma | complex.log_conj | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_inv_eq_ite (x : ℂ) : log (x⁻¹) = if x.arg = π then -conj (log x) else -log x | begin
by_cases hx : x = 0,
{ simp [hx] },
rw [inv_def, log_mul_of_real, real.log_inv, of_real_neg, ←sub_eq_neg_add, log_conj_eq_ite],
{ simp_rw [log, map_add, map_mul, conj_of_real, conj_I, norm_sq_eq_abs, real.log_pow,
nat.cast_two, of_real_mul, of_real_bit0, of_real_one, neg_add, mul_neg, two_mul, neg_n... | lemma | complex.log_inv_eq_ite | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"complex.norm_sq_pos",
"inv_pos",
"map_mul",
"map_ne_zero",
"mul_neg",
"nat.cast_two",
"real.log_inv",
"real.log_pow",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_inv (x : ℂ) (hx : x.arg ≠ π) : log (x⁻¹) = -log x | by rw [log_inv_eq_ite, if_neg hx] | lemma | complex.log_inv | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_pi_I_ne_zero : (2 * π * I : ℂ) ≠ 0 | by norm_num [real.pi_ne_zero, I_ne_zero] | lemma | complex.two_pi_I_ne_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"real.pi_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_eq_one_iff {x : ℂ} : exp x = 1 ↔ ∃ n : ℤ, x = n * ((2 * π) * I) | begin
split,
{ intro h,
rcases exists_unique_add_zsmul_mem_Ioc real.two_pi_pos x.im (-π) with ⟨n, hn, -⟩,
use -n,
rw [int.cast_neg, neg_mul, eq_neg_iff_add_eq_zero],
have : (x + n * (2 * π * I)).im ∈ Ioc (-π) π, by simpa [two_mul, mul_add] using hn,
rw [← log_exp this.1 this.2, exp_periodic.int_... | lemma | complex.exp_eq_one_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exists_unique_add_zsmul_mem_Ioc",
"exp",
"exp_zero",
"int.cast_neg",
"neg_mul",
"real.two_pi_pos",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_eq_exp_iff_exp_sub_eq_one {x y : ℂ} : exp x = exp y ↔ exp (x - y) = 1 | by rw [exp_sub, div_eq_one_iff_eq (exp_ne_zero _)] | lemma | complex.exp_eq_exp_iff_exp_sub_eq_one | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"div_eq_one_iff_eq",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_eq_exp_iff_exists_int {x y : ℂ} : exp x = exp y ↔ ∃ n : ℤ, x = y + n * ((2 * π) * I) | by simp only [exp_eq_exp_iff_exp_sub_eq_one, exp_eq_one_iff, sub_eq_iff_eq_add'] | lemma | complex.exp_eq_exp_iff_exists_int | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
countable_preimage_exp {s : set ℂ} : (exp ⁻¹' s).countable ↔ s.countable | begin
refine ⟨λ hs, _, λ hs, _⟩,
{ refine ((hs.image exp).insert 0).mono _,
rw [image_preimage_eq_inter_range, range_exp, ← diff_eq, ← union_singleton, diff_union_self],
exact subset_union_left _ _ },
{ rw ← bUnion_preimage_singleton,
refine hs.bUnion (λ z hz, _),
rcases em (∃ w, exp w = z) with ⟨... | lemma | complex.countable_preimage_exp | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"countable",
"em",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_nhds_within_im_neg_of_re_neg_of_im_zero
{z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
tendsto log (𝓝[{z : ℂ | z.im < 0}] z) (𝓝 $ real.log (abs z) - π * I) | begin
have := (continuous_of_real.continuous_at.comp_continuous_within_at
(continuous_abs.continuous_within_at.log _)).tendsto.add
(((continuous_of_real.tendsto _).comp $
tendsto_arg_nhds_within_im_neg_of_re_neg_of_im_zero hre him).mul tendsto_const_nhds),
convert this,
{ simp [sub_eq_add_neg] },
{ ... | lemma | complex.tendsto_log_nhds_within_im_neg_of_re_neg_of_im_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"lift",
"real.log",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_log_of_re_neg_of_im_zero
{z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
continuous_within_at log {z : ℂ | 0 ≤ z.im} z | begin
have := (continuous_of_real.continuous_at.comp_continuous_within_at
(continuous_abs.continuous_within_at.log _)).tendsto.add
((continuous_of_real.continuous_at.comp_continuous_within_at $
continuous_within_at_arg_of_re_neg_of_im_zero hre him).mul tendsto_const_nhds),
convert this,
{ lift z to ℝ ... | lemma | complex.continuous_within_at_log_of_re_neg_of_im_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous_within_at",
"lift",
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_log_nhds_within_im_nonneg_of_re_neg_of_im_zero
{z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
tendsto log (𝓝[{z : ℂ | 0 ≤ z.im}] z) (𝓝 $ real.log (abs z) + π * I) | by simpa only [log, arg_eq_pi_iff.2 ⟨hre, him⟩]
using (continuous_within_at_log_of_re_neg_of_im_zero hre him).tendsto | lemma | complex.tendsto_log_nhds_within_im_nonneg_of_re_neg_of_im_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"real.log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp_comap_re_at_bot : map exp (comap re at_bot) = 𝓝[≠] 0 | by rw [← comap_exp_nhds_zero, map_comap, range_exp, nhds_within] | lemma | complex.map_exp_comap_re_at_bot | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp",
"nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_exp_comap_re_at_top : map exp (comap re at_top) = comap abs at_top | begin
rw [← comap_exp_comap_abs_at_top, map_comap, range_exp, inf_eq_left, le_principal_iff],
exact eventually_ne_of_tendsto_norm_at_top tendsto_comap 0
end | lemma | complex.map_exp_comap_re_at_top | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"exp",
"inf_eq_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_clog {x : ℂ} (h : 0 < x.re ∨ x.im ≠ 0) :
continuous_at log x | begin
refine continuous_at.add _ _,
{ refine continuous_of_real.continuous_at.comp _,
refine (real.continuous_at_log _).comp complex.continuous_abs.continuous_at,
rw complex.abs.ne_zero_iff,
rintro rfl,
simpa using h },
{ have h_cont_mul : continuous (λ x : ℂ, x * I), from continuous_id'.mul conti... | lemma | continuous_at_clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous",
"continuous_at",
"continuous_const",
"real.continuous_at_log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.clog {l : filter α} {f : α → ℂ} {x : ℂ} (h : tendsto f l (𝓝 x))
(hx : 0 < x.re ∨ x.im ≠ 0) :
tendsto (λ t, log (f t)) l (𝓝 $ log x) | (continuous_at_clog hx).tendsto.comp h | lemma | filter.tendsto.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous_at_clog",
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.clog {f : α → ℂ} {x : α} (h₁ : continuous_at f x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
continuous_at (λ t, log (f t)) x | h₁.clog h₂ | lemma | continuous_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.clog {f : α → ℂ} {s : set α} {x : α} (h₁ : continuous_within_at f s x)
(h₂ : 0 < (f x).re ∨ (f x).im ≠ 0) :
continuous_within_at (λ t, log (f t)) s x | h₁.clog h₂ | lemma | continuous_within_at.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.clog {f : α → ℂ} {s : set α} (h₁ : continuous_on f s)
(h₂ : ∀ x ∈ s, 0 < (f x).re ∨ (f x).im ≠ 0) :
continuous_on (λ t, log (f t)) s | λ x hx, (h₁ x hx).clog (h₂ x hx) | lemma | continuous_on.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.clog {f : α → ℂ} (h₁ : continuous f) (h₂ : ∀ x, 0 < (f x).re ∨ (f x).im ≠ 0) :
continuous (λ t, log (f t)) | continuous_iff_continuous_at.2 $ λ x, h₁.continuous_at.clog (h₂ x) | lemma | continuous.clog | analysis.special_functions.complex | src/analysis/special_functions/complex/log.lean | [
"analysis.special_functions.complex.arg",
"analysis.special_functions.log.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_exp : is_open_map exp | open_map_of_strict_deriv has_strict_deriv_at_exp exp_ne_zero | lemma | complex.is_open_map_exp | analysis.special_functions.complex | src/analysis/special_functions/complex/log_deriv.lean | [
"analysis.calculus.inverse",
"analysis.special_functions.complex.log",
"analysis.special_functions.exp_deriv"
] | [
"exp",
"has_strict_deriv_at_exp",
"is_open_map",
"open_map_of_strict_deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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