nevanlinna/Nevanlinna/complexHarmonic.lean

import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Symmetric
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
import Nevanlinna.cauchyRiemann
import Nevanlinna.laplace
import Nevanlinna.partialDeriv

variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
variable {F₁ : Type*} [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [CompleteSpace F₁]
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace ℝ G]
variable {G₁ : Type*} [NormedAddCommGroup G₁] [NormedSpace ℂ G₁] [CompleteSpace G₁]


def Harmonic (f : ℂ → F) : Prop :=
  (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)


def HarmonicOn (f : ℂ → F) (s : Set ℂ) : Prop :=
  (ContDiffOn ℝ 2 f s) ∧ (∀ z ∈ s, Complex.laplace f z = 0)


theorem HarmonicOn_of_locally_HarmonicOn {f : ℂ → F} {s : Set ℂ} (h : ∀ x ∈ s, ∃ (u : Set ℂ), IsOpen u ∧ x ∈ u ∧ HarmonicOn f (s ∩ u)) :
  HarmonicOn f s := by
  constructor
  · apply contDiffOn_of_locally_contDiffOn
    intro x xHyp
    obtain ⟨u, uHyp⟩ := h x xHyp
    use u
    exact ⟨ uHyp.1, ⟨uHyp.2.1, uHyp.2.2.1⟩⟩
  · intro x xHyp
    obtain ⟨u, uHyp⟩ := h x xHyp
    exact (uHyp.2.2.2) x ⟨xHyp, uHyp.2.1⟩


theorem HarmonicOn_congr {f₁ f₂ : ℂ → F} {s : Set ℂ} (hs : IsOpen s) (hf₁₂ : ∀ x ∈ s, f₁ x = f₂ x) :
  HarmonicOn f₁ s ↔ HarmonicOn f₂ s := by
  constructor
  · intro h₁
    constructor
    · apply ContDiffOn.congr h₁.1
      intro x hx
      rw [eq_comm]
      exact hf₁₂ x hx
    · intro z hz
      have : f₁ =ᶠ[nhds z] f₂ := by
        unfold Filter.EventuallyEq
        unfold Filter.Eventually
        simp
        refine mem_nhds_iff.mpr ?_
        use s
        constructor
        · exact hf₁₂
        · constructor
          · exact hs
          · exact hz
      rw [← laplace_eventuallyEq this]
      exact h₁.2 z hz
  · intro h₁
    constructor
    · apply ContDiffOn.congr h₁.1
      intro x hx
      exact hf₁₂ x hx
    · intro z hz
      have : f₁ =ᶠ[nhds z] f₂ := by
        unfold Filter.EventuallyEq
        unfold Filter.Eventually
        simp
        refine mem_nhds_iff.mpr ?_
        use s
        constructor
        · exact hf₁₂
        · constructor
          · exact hs
          · exact hz
      rw [laplace_eventuallyEq this]
      exact h₁.2 z hz


theorem harmonic_add_harmonic_is_harmonic {f₁ f₂ : ℂ → F} (h₁ : Harmonic f₁) (h₂ : Harmonic f₂) :
  Harmonic (f₁ + f₂) := by
  constructor
  · exact ContDiff.add h₁.1 h₂.1
  · rw [laplace_add h₁.1 h₂.1]
    simp
    intro z
    rw [h₁.2 z, h₂.2 z]
    simp


theorem harmonicOn_add_harmonicOn_is_harmonicOn {f₁ f₂ : ℂ → F} {s : Set ℂ} (hs : IsOpen s) (h₁ : HarmonicOn f₁ s) (h₂ : HarmonicOn f₂ s) :
  HarmonicOn (f₁ + f₂) s := by
  constructor
  · exact ContDiffOn.add h₁.1 h₂.1
  · intro z hz
    rw [laplace_add_ContDiffOn hs h₁.1 h₂.1 z hz]
    rw [h₁.2 z hz, h₂.2 z hz]
    simp


theorem harmonic_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (h : Harmonic f) :
  Harmonic (c • f) := by
  constructor
  · exact ContDiff.const_smul c h.1
  · rw [laplace_smul]
    dsimp
    intro z
    rw [h.2 z]
    simp


theorem harmonic_iff_smul_const_is_harmonic {f : ℂ → F} {c : ℝ} (hc : c ≠ 0) :
  Harmonic f ↔ Harmonic (c • f) := by
  constructor
  · exact harmonic_smul_const_is_harmonic
  · nth_rewrite 2 [((eq_inv_smul_iff₀ hc).mpr rfl : f = c⁻¹ • c • f)]
    exact fun a => harmonic_smul_const_is_harmonic a


theorem harmonic_comp_CLM_is_harmonic {f : ℂ → F₁} {l : F₁ →L[ℝ] G} (h : Harmonic f) :
  Harmonic (l ∘ f) := by

  constructor
  · -- Continuous differentiability
    apply ContDiff.comp
    exact ContinuousLinearMap.contDiff l
    exact h.1
  · rw [laplace_compContLin]
    simp
    intro z
    rw [h.2 z]
    simp
    exact ContDiff.restrict_scalars ℝ h.1


theorem harmonicOn_comp_CLM_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l : F₁ →L[ℝ] G} (hs : IsOpen s) (h : HarmonicOn f s) :
  HarmonicOn (l ∘ f) s := by

  constructor
  · -- Continuous differentiability
    apply ContDiffOn.continuousLinearMap_comp
    exact h.1
  · -- Vanishing of Laplace
    intro z zHyp
    rw [laplace_compContLinAt]
    simp
    rw [h.2 z]
    simp
    assumption
    apply ContDiffOn.contDiffAt h.1
    exact IsOpen.mem_nhds hs zHyp


theorem harmonic_iff_comp_CLE_is_harmonic {f : ℂ → F₁} {l : F₁ ≃L[ℝ] G₁} :
  Harmonic f ↔ Harmonic (l ∘ f) := by

  constructor
  · have : l ∘ f = (l : F₁ →L[ℝ] G₁) ∘ f := by rfl
    rw [this]
    exact harmonic_comp_CLM_is_harmonic
  · have : f = (l.symm : G₁ →L[ℝ] F₁) ∘ l ∘ f := by
      funext z
      unfold Function.comp
      simp
    nth_rewrite 2 [this]
    exact harmonic_comp_CLM_is_harmonic


theorem harmonicOn_iff_comp_CLE_is_harmonicOn {f : ℂ → F₁} {s : Set ℂ} {l : F₁ ≃L[ℝ] G₁} (hs : IsOpen s) :
  HarmonicOn f s ↔ HarmonicOn (l ∘ f) s := by

  constructor
  · have : l ∘ f = (l : F₁ →L[ℝ] G₁) ∘ f := by rfl
    rw [this]
    exact harmonicOn_comp_CLM_is_harmonicOn hs
  · have : f = (l.symm : G₁ →L[ℝ] F₁) ∘ l ∘ f := by
      funext z
      unfold Function.comp
      simp
    nth_rewrite 2 [this]
    exact harmonicOn_comp_CLM_is_harmonicOn hs