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