nevanlinna/Nevanlinna/complexHarmonic.lean

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2024-04-30 08:20:57 +02:00
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Nevanlinna.cauchyRiemann
noncomputable def Complex.laplace : () → () := by
intro f
let f₁ := fun x ↦ lineDeriv f x 1
let f₁₁ := fun x ↦ lineDeriv f₁ x 1
let f₂ := fun x ↦ lineDeriv f x Complex.I
let f₂₂ := fun x ↦ lineDeriv f₂ x Complex.I
exact f₁₁ + f₂₂
def Harmonic (f : ) : Prop :=
(ContDiff 2 f) ∧ (∀ z, Complex.laplace f z = 0)
theorem re_comp_holomorphic_is_harmonic (f : ) :
Differentiable f → Harmonic (Complex.reCLM ∘ f) := by
intro h
constructor
· -- f is two times real continuously differentiable
sorry
· -- Laplace of f is zero
intro z
unfold Complex.laplace
simp
let ZZ := (CauchyRiemann₃ (h z)).left
rw [ZZ]
sorry