nevanlinna/Nevanlinna/holomorphic.lean

import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Nevanlinna.partialDeriv

variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]


def HolomorphicAt (f : E → F) (x : E) : Prop :=
  ∃ s ∈ nhds x, ∀ z ∈ s, DifferentiableAt ℂ f z


theorem HolomorphicAt_iff
  {f : E → F}
  {x : E} :
  HolomorphicAt f x ↔ ∃ s : Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, DifferentiableAt ℂ f z) := by
  constructor
  · intro hf
    obtain ⟨t, h₁t, h₂t⟩ := hf
    obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
    use s
    constructor
    · assumption
    · constructor
      · assumption
      · intro z hz
        exact h₂t z (h₁s hz)
  · intro hyp
    obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
    use s
    constructor
    · apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
    · assumption


theorem HolomorphicAt_isOpen
  {f : E → F}
  {x : E} :
  HolomorphicAt f x ↔ ∃ s : Set E, IsOpen s ∧ x ∈ s ∧ (∀ z ∈ s, HolomorphicAt f z) := by
  constructor
  · intro hf
    obtain ⟨t, h₁t, h₂t⟩ := hf
    obtain ⟨s, h₁s, h₂s, h₃s⟩ := mem_nhds_iff.1 h₁t
    use s
    constructor
    · assumption
    · constructor
      · assumption
      · intro z hz
        apply HolomorphicAt_iff.2
        use s
        constructor
        · assumption
        · constructor
          · assumption
          · exact fun w hw ↦ h₂t w (h₁s hw)
  · intro hyp
    obtain ⟨s, h₁s, h₂s, hf⟩ := hyp
    use s
    constructor
    · apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
    · intro z hz
      obtain ⟨t, ht⟩ := (hf z hz)
      exact ht.2 z (mem_of_mem_nhds ht.1)


theorem CauchyRiemann'₅
  {f : ℂ → F}
  {z : ℂ}
  (h : DifferentiableAt ℂ f z) :
  partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
  unfold partialDeriv

  conv =>
    left
    rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
    simp
    rw [← mul_one Complex.I]
    rw [← smul_eq_mul]
    rw [ContinuousLinearMap.map_smul_of_tower (fderiv ℂ f z) Complex.I 1]
  conv =>
    right
    right
    rw [DifferentiableAt.fderiv_restrictScalars ℝ h]


theorem CauchyRiemann'₆
  {f : ℂ → F}
  {z : ℂ}
  (h : HolomorphicAt f z) :
  partialDeriv ℝ Complex.I f =ᶠ[nhds z] Complex.I • partialDeriv ℝ 1 f := by

  obtain ⟨s, h₁s, hz, h₂f⟩ := HolomorphicAt_iff.1 h

  apply Filter.eventuallyEq_iff_exists_mem.2
  use s
  constructor
  · exact IsOpen.mem_nhds h₁s hz
  · intro w hw
    apply CauchyRiemann'₅
    exact h₂f w hw