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Update complexHarmonic.lean

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Stefan Kebekus 2024-05-13 10:10:10 +02:00
parent 5e3f9c463f
commit e7b23a6b2c
1 changed files with 26 additions and 1 deletions
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Nevanlinna/complexHarmonic.lean
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@ -85,5 +85,30 @@ theorem holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
theorem re_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
Harmonic (Complex.reCLM ∘ f) := by
constructor
· -- Continuous differentiability
apply ContDiff.comp
exact ContinuousLinearMap.contDiff Complex.reCLM
exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
· rw [laplace_compContLin]
simp
intro z
rw [(holomorphic_is_harmonic h).right z]
simp
exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
sorry
theorem im_of_holomorphic_is_harmonic {f : ℂ → ℂ} (h : Differentiable ℂ f) :
Harmonic (Complex.imCLM ∘ f) := by
constructor
· -- Continuous differentiability
apply ContDiff.comp
exact ContinuousLinearMap.contDiff Complex.imCLM
exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)
· rw [laplace_compContLin]
simp
intro z
rw [(holomorphic_is_harmonic h).right z]
simp
exact ContDiff.restrict_scalars ℝ (Differentiable.contDiff h)