feat: port Analysis.SpecialFunctions.Trigonometric.ComplexDeriv (#4753)

Mathlib.lean

@@ -656,6 +656,7 @@ import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
 import Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv

Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean

@@ -0,0 +1,82 @@
+/-
+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
+
+! This file was ported from Lean 3 source module analysis.special_functions.trigonometric.complex_deriv
+! leanprover-community/mathlib commit 2c1d8ca2812b64f88992a5294ea3dba144755cd1
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
+
+/-!
+# Complex trigonometric functions
+
+Basic facts and derivatives for the complex trigonometric functions.
+-/
+
+
+noncomputable section
+
+local macro_rules | `($x ^ $y) => `(HPow.hPow $x $y) -- Porting note: See issue #2220
+
+namespace Complex
+
+open Set Filter
+
+open scoped Real
+
+theorem hasStrictDerivAt_tan {x : ℂ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x := by
+  convert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1
+  rw_mod_cast [← sin_sq_add_cos_sq x]
+  ring
+#align complex.has_strict_deriv_at_tan Complex.hasStrictDerivAt_tan
+
+theorem hasDerivAt_tan {x : ℂ} (h : cos x ≠ 0) : HasDerivAt tan (1 / cos x ^ 2) x :=
+  (hasStrictDerivAt_tan h).hasDerivAt
+#align complex.has_deriv_at_tan Complex.hasDerivAt_tan
+
+open scoped Topology
+
+theorem tendsto_abs_tan_of_cos_eq_zero {x : ℂ} (hx : cos x = 0) :
+    Tendsto (fun x => abs (tan x)) (𝓝[≠] x) atTop := by
+  simp only [tan_eq_sin_div_cos, ← norm_eq_abs, norm_div]
+  have A : sin x ≠ 0 := fun h => by simpa [*, sq] using sin_sq_add_cos_sq x
+  have B : Tendsto cos (𝓝[≠] x) (𝓝[≠] 0) :=
+    hx ▸ (hasDerivAt_cos x).tendsto_punctured_nhds (neg_ne_zero.2 A)
+  exact continuous_sin.continuousWithinAt.norm.mul_atTop (norm_pos_iff.2 A)
+    (tendsto_norm_nhdsWithin_zero.comp B).inv_tendsto_zero
+#align complex.tendsto_abs_tan_of_cos_eq_zero Complex.tendsto_abs_tan_of_cos_eq_zero
+
+theorem tendsto_abs_tan_atTop (k : ℤ) :
+    Tendsto (fun x => abs (tan x)) (𝓝[≠] ((2 * k + 1) * π / 2 : ℂ)) atTop :=
+  tendsto_abs_tan_of_cos_eq_zero <| cos_eq_zero_iff.2 ⟨k, rfl⟩
+#align complex.tendsto_abs_tan_at_top Complex.tendsto_abs_tan_atTop
+
+@[simp]
+theorem continuousAt_tan {x : ℂ} : ContinuousAt tan x ↔ cos x ≠ 0 := by
+  refine' ⟨fun hc h₀ => _, fun h => (hasDerivAt_tan h).continuousAt⟩
+  exact not_tendsto_nhds_of_tendsto_atTop (tendsto_abs_tan_of_cos_eq_zero h₀) _
+    (hc.norm.tendsto.mono_left inf_le_left)
+#align complex.continuous_at_tan Complex.continuousAt_tan
+
+@[simp]
+theorem differentiableAt_tan {x : ℂ} : DifferentiableAt ℂ tan x ↔ cos x ≠ 0 :=
+  ⟨fun h => continuousAt_tan.1 h.continuousAt, fun h => (hasDerivAt_tan h).differentiableAt⟩
+#align complex.differentiable_at_tan Complex.differentiableAt_tan
+
+@[simp]
+theorem deriv_tan (x : ℂ) : deriv tan x = 1 / cos x ^ 2 :=
+  if h : cos x = 0 then by
+    have : ¬DifferentiableAt ℂ tan x := mt differentiableAt_tan.1 (Classical.not_not.2 h)
+    simp [deriv_zero_of_not_differentiableAt this, h, sq]
+  else (hasDerivAt_tan h).deriv
+#align complex.deriv_tan Complex.deriv_tan
+
+@[simp]
+theorem contDiffAt_tan {x : ℂ} {n : ℕ∞} : ContDiffAt ℂ n tan x ↔ cos x ≠ 0 :=
+  ⟨fun h => continuousAt_tan.1 h.continuousAt, contDiff_sin.contDiffAt.div contDiff_cos.contDiffAt⟩
+#align complex.cont_diff_at_tan Complex.contDiffAt_tan
+
+end Complex