nevanlinna/Nevanlinna/holomorphic.examples.lean

import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Nevanlinna.complexHarmonic
import Nevanlinna.complexHarmonic
import Nevanlinna.holomorphicAt


theorem CauchyRiemann₆
  {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] 
  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] 
  {f : E → F}
  {z : E} : 
  (DifferentiableAt ℂ f z) ↔ (DifferentiableAt ℝ f z) ∧ ∀ e, partialDeriv ℝ (Complex.I • e) f z = Complex.I • partialDeriv ℝ e f z := by
  constructor

  · -- Direction "→" 
    intro h
    constructor
    · exact DifferentiableAt.restrictScalars ℝ h
    · unfold partialDeriv
      conv =>
        intro e
        left
        rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
        simp
        rw [← mul_one Complex.I]
        rw [← smul_eq_mul]
      conv =>
        intro e
        right
        right
        rw [DifferentiableAt.fderiv_restrictScalars ℝ h]
      simp

  · -- Direction "←" 
    intro ⟨h₁, h₂⟩
    apply (differentiableAt_iff_restrictScalars ℝ h₁).2

    use {
      toFun := fderiv ℝ f z
      map_add' := fun x y => ContinuousLinearMap.map_add (fderiv ℝ f z) x y
      map_smul' := by
        simp
        intro m x
        have : m = m.re + m.im • Complex.I := by simp
        rw [this, add_smul, add_smul, ContinuousLinearMap.map_add]
        congr
        simp
        rw [smul_assoc, smul_assoc, ContinuousLinearMap.map_smul (fderiv ℝ f z) m.2]
        congr
        exact h₂ x
    }
    rfl


theorem CauchyRiemann₇
  {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F] 
  {f : ℂ → F}
  {z : ℂ} : 
  (DifferentiableAt ℂ f z) ↔ (DifferentiableAt ℝ f z) ∧ partialDeriv ℝ Complex.I f z = Complex.I • partialDeriv ℝ 1 f z := by
  constructor
  · intro hf
    constructor
    · exact (CauchyRiemann₆.1 hf).1
    · let A := (CauchyRiemann₆.1 hf).2 1
      simp at A
      exact A
  · intro ⟨h₁, h₂⟩
    apply CauchyRiemann₆.2
    constructor
    · exact h₁
    · intro e
      have : Complex.I • e = e • Complex.I := by 
        rw [smul_eq_mul, smul_eq_mul]
        exact CommMonoid.mul_comm Complex.I e
      rw [this]
      have : e = e.re + e.im • Complex.I := by simp
      rw [this, add_smul, partialDeriv_add₁, partialDeriv_add₁]
      simp
      rw [← smul_eq_mul]
      have : partialDeriv ℝ ((e.re : ℝ) • Complex.I) f = partialDeriv ℝ ((e.re : ℂ) • Complex.I) f := by rfl
      rw [← this, partialDeriv_smul₁ ℝ]
      have : (e.re : ℂ) = (e.re : ℝ) • (1 : ℂ) := by simp
      rw [this, partialDeriv_smul₁ ℝ]
      have : partialDeriv ℝ ((e.im : ℂ) * Complex.I) f = partialDeriv ℝ ((e.im : ℝ) • Complex.I) f := by rfl
      rw [this, partialDeriv_smul₁ ℝ]
      simp
      rw [h₂]
      rw [smul_comm]
      congr
      rw [mul_assoc]
      simp
      nth_rw 2 [smul_comm]
      rw [← smul_assoc]
      simp
      have : - (e.im : ℂ) = (-e.im : ℝ) • (1 : ℂ) := by simp
      rw [this, partialDeriv_smul₁ ℝ]
      simp
  


/-

A harmonic, real-valued function on ℂ is the real part of a suitable holomorphic function.

-/

theorem harmonic_is_realOfHolomorphic
  {f : ℂ → ℝ}
  (hf : ∀ z, HarmonicAt f z) :
  ∃ F : ℂ → ℂ, ∀ z, (HolomorphicAt F z ∧ ((F z).re = f z)) := by

  let f_1 : ℂ → ℂ := Complex.ofRealCLM ∘ (partialDeriv ℝ 1 f)
  let f_I : ℂ → ℂ := Complex.ofRealCLM ∘ (partialDeriv ℝ Complex.I f)

  let g : ℂ → ℂ := f_1 - Complex.I • f_I

  have reg₀ : Differentiable ℝ g := by 
    let smulICLM : ℂ ≃L[ℝ] ℂ :=
    {
      toFun := fun x ↦ Complex.I • x
      map_add' := fun x y => DistribSMul.smul_add Complex.I x y
      map_smul' := fun m x => (smul_comm ((RingHom.id ℝ) m) Complex.I x).symm
      invFun := fun x ↦ (Complex.I)⁻¹ • x
      left_inv := by
        intro x
        simp
        rw [← mul_assoc, mul_comm]
        simp
      right_inv := by
        intro x
        simp
        rw [← mul_assoc]
        simp
      continuous_toFun := continuous_const_smul Complex.I
      continuous_invFun := continuous_const_smul (Complex.I)⁻¹
    }

    apply Differentiable.sub
    apply Differentiable.comp
    exact ContinuousLinearMap.differentiable Complex.ofRealCLM
    intro z
    sorry
    apply Differentiable.comp
    sorry
    sorry

  
  have reg₁ : Differentiable ℂ g := by 
    intro z
    apply CauchyRiemann₇.2
    constructor
    · exact reg₀ z
    · dsimp [g]
      have : f_1 - Complex.I • f_I = f_1 + (- Complex.I • f_I) := by
        rw [sub_eq_add_neg]
        simp
      rw [this, partialDeriv_add₂, partialDeriv_add₂]
      simp
      dsimp [f_1, f_I]
      
      sorry
      sorry
      sorry
      sorry
      sorry

  


  sorry