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Update complexHarmonic.lean

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Stefan Kebekus 2024-05-02 09:48:26 +02:00
parent f85fafd05f
commit b9a973d10d
1 changed files with 21 additions and 6 deletions
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Nevanlinna/complexHarmonic.lean
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@ -1,4 +1,5 @@
import Mathlib.Analysis.Complex.Basic import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Analysis.Calculus.LineDeriv.Basic import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Basic import Mathlib.Analysis.Calculus.FDeriv.Basic
@ -16,22 +17,36 @@ noncomputable def Complex.laplace : (ℂ → ℝ) → (ℂ → ℝ) := by
def Harmonic (f : ℂ → ℝ) : Prop := def Harmonic (f : ℂ → ℝ) : Prop :=
(ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0) (ContDiff ℝ 2 f) ∧ (∀ z, Complex.laplace f z = 0)
#check contDiff_iff_ftaylorSeries.2
theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) : lemma c2_if_holomorphic (f : ℂ → ℂ) : Differentiable ℂ f → ContDiff ℂ 2 f := by
intro fHyp
exact Differentiable.contDiff fHyp
lemma c2R_if_holomorphic (f : ℂ → ℂ) : Differentiable ℂ f → ContDiff ℝ 2 f := by
intro fHyp
let ZZ := c2_if_holomorphic f fHyp
apply ContDiff.restrict_scalars ℝ ZZ
theorem re_comp_holomorphic_is_harmonic (f : ℂ → ℂ) :
Differentiable ℂ f → Harmonic (Complex.reCLM ∘ f) := by Differentiable ℂ f → Harmonic (Complex.reCLM ∘ f) := by
intro h intro h
constructor constructor
· -- f is two times real continuously differentiable · -- Complex.reCLM ∘ f is two times real continuously differentiable
sorry apply ContDiff.comp
· -- Complex.reCLM is two times real continuously differentiable
exact ContinuousLinearMap.contDiff Complex.reCLM
· -- f is two times real continuously differentiable
exact c2R_if_holomorphic f h
· -- Laplace of f is zero · -- Laplace of f is zero
intro z intro z
unfold Complex.laplace unfold Complex.laplace
simp simp
let ZZ := (CauchyRiemann₃ (h z)).left let ZZ := (CauchyRiemann₃ (h z)).left
rw [ZZ]
sorry
sorry