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Stefan Kebekus 2024-06-07 10:28:11 +02:00
parent 2f4672e144
commit 271ad821dd
2 changed files with 120 additions and 3 deletions

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@ -60,7 +60,7 @@ theorem holomorphicAt_is_harmonicAt
theorem re_of_holomorphicAt_is_harmonicAr
{f : }
{f : }
{z : }
(h : HolomorphicAt f z) :
HarmonicAt (Complex.reCLM ∘ f) z := by
@ -86,6 +86,123 @@ theorem antiholomorphicAt_is_harmonicAt
exact holomorphicAt_is_harmonicAt h
theorem log_normSq_of_holomorphicAt_is_harmonicAt
{f : }
{z : }
(h₁f : HolomorphicAt f z)
(h₂f : f z ≠ 0) :
HarmonicAt (Real.log ∘ Complex.normSq ∘ f) z := by
/- First prove the theorem under the additional assumption that
-/
have lem₁ : ∀ g : , (HolomorphicAt g z) → (g z ≠ 0) → (g z ∈ Complex.slitPlane) → HarmonicAt (Real.log ∘ Complex.normSq ∘ g) z := by
intro g h₁g h₂g h₃g
-- Rewrite the log |g|² as Complex.log (g * gc)
suffices hyp : HarmonicAt (Complex.log ∘ ((Complex.conjCLE ∘ g) * g)) z from by
have : Real.log ∘ Complex.normSq ∘ g = Complex.reCLM ∘ Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ g := by
funext x
simp
rw [this]
have : Complex.ofRealCLM ∘ Real.log ∘ Complex.normSq ∘ g = Complex.log ∘ ((Complex.conjCLE ∘ g) * g) := by
funext x
simp
rw [Complex.ofReal_log]
rw [Complex.normSq_eq_conj_mul_self]
exact Complex.normSq_nonneg (g x)
rw [← this] at hyp
apply harmonicAt_comp_CLM_is_harmonicAt hyp
-- Locally around z, rewrite Complex.log (g * gc) as Complex.log g + Complex.log.gc
-- This uses the assumption that g z is in Complex.slitPlane
have : (Complex.log ∘ (Complex.conjCLE ∘ g * g)) =ᶠ[nhds z] (Complex.log ∘ Complex.conjCLE ∘ g + Complex.log ∘ g) := by
apply Filter.eventuallyEq_iff_exists_mem.2
use g⁻¹' (Complex.slitPlane ∩ {0}ᶜ)
constructor
· apply ContinuousAt.preimage_mem_nhds
· exact (HolomorphicAt_differentiableAt h₁g).continuousAt
· apply IsOpen.mem_nhds
apply IsOpen.inter Complex.isOpen_slitPlane isOpen_ne
constructor
· exact h₃g
· exact h₂g
· intro x hx
simp
rw [Complex.log_mul_eq_add_log_iff _ hx.2]
rw [Complex.arg_conj]
simp [Complex.slitPlane_arg_ne_pi hx.1]
constructor
· exact Real.pi_pos
· exact Real.pi_nonneg
simp
apply hx.2
sorry
-- Suffices to show Harmonic (Complex.log ∘ ⇑(starRingEnd ) ∘ f + Complex.log ∘ f)
-- THIS IS WHERE WE USE h₃
have : (Complex.log ∘ (Complex.conjCLE ∘ f * f)) z = (Complex.log ∘ Complex.conjCLE ∘ f + Complex.log ∘ f) z := by
unfold Function.comp
simp
rw [Complex.log_mul_eq_add_log_iff]
have : Complex.arg ((starRingEnd ) (f z)) = - Complex.arg (f z) := by
rw [Complex.arg_conj]
have : ¬ Complex.arg (f z) = Real.pi := by
exact Complex.slitPlane_arg_ne_pi (h₃ z hz)
simp
tauto
rw [this]
simp
constructor
· exact Real.pi_pos
· exact Real.pi_nonneg
exact (AddEquivClass.map_ne_zero_iff starRingAut).mpr (h₂ z hz)
exact h₂ z hz
rw [HarmonicOn_congr hs this]
simp
apply harmonicOn_add_harmonicOn_is_harmonicOn hs
have : (fun x => Complex.log ((starRingEnd ) (f x))) = (Complex.log ∘ ⇑(starRingEnd ) ∘ f) := by
rfl
rw [this]
-- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ) ∘ f) s
have : ∀ z ∈ s, (Complex.log ∘ ⇑(starRingEnd ) ∘ f) z = (Complex.conjCLE ∘ Complex.log ∘ f) z := by
intro z hz
unfold Function.comp
rw [Complex.log_conj]
rfl
exact Complex.slitPlane_arg_ne_pi (h₃ z hz)
rw [HarmonicOn_congr hs this]
rw [← harmonicOn_iff_comp_CLE_is_harmonicOn]
apply holomorphicOn_is_harmonicOn
exact hs
intro z hz
apply DifferentiableAt.differentiableWithinAt
apply DifferentiableAt.comp
exact Complex.differentiableAt_log (h₃ z hz)
apply DifferentiableOn.differentiableAt h₁ -- (h₁ z hz)
exact IsOpen.mem_nhds hs hz
exact hs
-- HarmonicOn (Complex.log ∘ ⇑(starRingEnd ) ∘ f) s
apply holomorphicOn_is_harmonicOn hs
exact DifferentiableOn.clog h₁ h₃
theorem holomorphic_is_harmonic {f : → F₁} (h : Differentiable f) :
Harmonic f := by

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@ -243,8 +243,8 @@ theorem harmonic_iff_comp_CLE_is_harmonic {f : → F₁} {l : F₁ ≃L[]
exact harmonic_comp_CLM_is_harmonic
theorem harmonicAt_iff_comp_CLE_is_harmonicAt
{f : → F₁}
theorem harmonicAt_iff_comp_CLE_is_harmonicAt
{f : → F₁}
{z : }
{l : F₁ ≃L[] G₁} :
HarmonicAt f z ↔ HarmonicAt (l ∘ f) z := by