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Update complexHarmonic.lean

This commit is contained in:
Stefan Kebekus 2024-05-16 10:16:29 +02:00
parent 710de9372b
commit 077bace964
1 changed files with 2 additions and 12 deletions
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Nevanlinna/complexHarmonic.lean
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@ -162,9 +162,6 @@ theorem logabs_of_holomorphic_is_harmonic
apply ContDiff.comp_contDiffAt z normSq_is_real_C2 apply ContDiff.comp_contDiffAt z normSq_is_real_C2
exact ContDiff.contDiffAt f_is_real_C2 exact ContDiff.contDiffAt f_is_real_C2
have t₂ : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by have t₂ : Complex.log ∘ ⇑(starRingEnd ℂ) ∘ f = Complex.conjCLE ∘ Complex.log ∘ f := by
funext z funext z
unfold Function.comp unfold Function.comp
@ -172,9 +169,6 @@ theorem logabs_of_holomorphic_is_harmonic
rfl rfl
exact Complex.slitPlane_arg_ne_pi (h₃ z) exact Complex.slitPlane_arg_ne_pi (h₃ z)
constructor constructor
· -- logabs f is real C² · -- logabs f is real C²
have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by
@ -188,14 +182,10 @@ theorem logabs_of_holomorphic_is_harmonic
rw [this] rw [this]
have : (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) = (fun z ↦ (2 : ℝ)⁻¹ • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by have : (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) = (fun z ↦ (2 : ℝ)⁻¹ • ((Real.log ∘ ⇑Complex.normSq ∘ f) z)) := by
simp
exact rfl exact rfl
rw [this] rw [this]
apply ContDiff.const_smul
apply contDiff_iff_contDiffAt.2 exact t₄
intro z
apply ContDiffAt.const_smul
exact ContDiff.contDiffAt t₄
· -- Laplace vanishes · -- Laplace vanishes
have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by have : (fun z ↦ Real.log ‖f z‖) = (2 : ℝ)⁻¹ • (Real.log ∘ Complex.normSq ∘ f) := by