kebekus
/
nevanlinna
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

working…

This commit is contained in:
Stefan Kebekus 2024-08-08 14:26:53 +02:00
parent 75ce3b31ef
commit 4642c017c7
2 changed files with 40 additions and 17 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

55
Nevanlinna/harmonicAt_meanValue.lean
View File

@ -3,28 +3,51 @@ import Nevanlinna.holomorphic_examples
theorem harmonic_meanValue theorem harmonic_meanValue
{f : ℂ → ℝ} {f : ℂ → ℝ}
(hf : ∀ z, HarmonicAt f z) {z : ℂ}
(R : ℝ) (ρ R : ℝ)
(hR : R > 0) : (hR : R > 0)
(∫ (x : ℝ) in (0)..2 * Real.pi, f (circleMap 0 R x)) = 2 * Real.pi * f 0 (hρ : ρ > R)
(hf : ∀ x ∈ Metric.ball z ρ , HarmonicAt f x)
:
(∫ (x : ℝ) in (0)..2 * Real.pi, f (circleMap z R x)) = 2 * Real.pi * f z
:= by := by
obtain ⟨F, h₁F, h₂F⟩ := harmonic_is_realOfHolomorphic hf obtain ⟨F, h₁F, h₂F⟩ := harmonic_is_realOfHolomorphic (gt_trans hρ hR) hf
have regF : Differentiable ℂ F := fun z ↦ HolomorphicAt.differentiableAt (h₁F z) have hrρ : Metric.ball z R ⊆ Metric.ball z ρ := by
intro x hx
exact gt_trans hρ hx
have : (∮ (z : ℂ) in C(0, R), z⁻¹ • F z) = (2 * ↑Real.pi * Complex.I) • F 0 := by have reg₀F : DifferentiableOn ℂ F (Metric.ball z ρ) := by
intro x hx
apply DifferentiableAt.differentiableWithinAt
apply HolomorphicAt.differentiableAt (h₁F x _)
exact hx
have reg₁F : DifferentiableOn ℂ F (Metric.ball z R) := by
intro x hx
apply DifferentiableAt.differentiableWithinAt
apply HolomorphicAt.differentiableAt (h₁F x _)
exact hrρ hx
have : (∮ (x : ℂ) in C(z, R), (x - z)⁻¹ • F x) = (2 * ↑Real.pi * Complex.I) • F z := by
let s : Set ℂ := ∅ let s : Set ℂ := ∅
let hs : s.Countable := Set.countable_empty let hs : s.Countable := Set.countable_empty
let _ : ℂ := 0 let _ : ℂ := 0
let hw : (0 : ℂ) ∈ Metric.ball 0 R := Metric.mem_ball_self hR have hw : (z : ℂ) ∈ Metric.ball z R := Metric.mem_ball_self hR
let hc : ContinuousOn F (Metric.closedBall 0 R) := by have hc : ContinuousOn F (Metric.closedBall z R) := by
apply Continuous.continuousOn apply reg₀F.continuousOn.mono
exact regF.continuous intro x hx
let hd : ∀ x ∈ Metric.ball 0 R \ s, DifferentiableAt ℂ F x := by simp at hx
intro x _ simp
exact regF x linarith
have hd : ∀ x ∈ Metric.ball z R \ s, DifferentiableAt ℂ F x := by
intro x hx
let A := reg₁F x hx.1
apply A.differentiableAt
apply (IsOpen.mem_nhds_iff ?hs).mpr
exact hx.1
exact Metric.isOpen_ball
let CIF := Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable hs hw hc hd let CIF := Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable hs hw hc hd
simp at CIF simp at CIF
assumption assumption
@ -32,7 +55,7 @@ theorem harmonic_meanValue
unfold circleIntegral at this unfold circleIntegral at this
simp_rw [deriv_circleMap] at this simp_rw [deriv_circleMap] at this
have t₁ {θ : ℝ} : (circleMap 0 R θ * Complex.I) • (circleMap 0 R θ)⁻¹ • F (circleMap 0 R θ) = Complex.I • F (circleMap 0 R θ) := by have t₁ {θ : ℝ} : (circleMap 0 R θ * Complex.I) • (circleMap 0 R θ)⁻¹ • F (circleMap z R θ) = Complex.I • F (circleMap z R θ) := by
rw [← smul_assoc] rw [← smul_assoc]
congr 1 congr 1
simp simp

2
Nevanlinna/holomorphic_examples.lean
View File

@ -1,6 +1,6 @@
import Nevanlinna.complexHarmonic import Nevanlinna.complexHarmonic
import Nevanlinna.holomorphicAt import Nevanlinna.holomorphicAt
import Nevanlinna.holomorphic_primitive2 import Nevanlinna.holomorphic_primitive
import Nevanlinna.mathlibAddOn import Nevanlinna.mathlibAddOn