108 lines
3.4 KiB
Plaintext
108 lines
3.4 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
|
||
import Nevanlinna.holomorphic_examples
|
||
|
||
theorem harmonic_meanValue
|
||
{f : ℂ → ℝ}
|
||
{z : ℂ}
|
||
(ρ R : ℝ)
|
||
(hR : R > 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
|
||
|
||
obtain ⟨F, h₁F, h₂F⟩ := harmonic_is_realOfHolomorphic (gt_trans hρ hR) hf
|
||
|
||
have hrρ : Metric.ball z R ⊆ Metric.ball z ρ := by
|
||
intro x hx
|
||
exact gt_trans hρ hx
|
||
|
||
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 hs : s.Countable := Set.countable_empty
|
||
let _ : ℂ := 0
|
||
have hw : (z : ℂ) ∈ Metric.ball z R := Metric.mem_ball_self hR
|
||
have hc : ContinuousOn F (Metric.closedBall z R) := by
|
||
apply reg₀F.continuousOn.mono
|
||
intro x hx
|
||
simp at hx
|
||
simp
|
||
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
|
||
simp at CIF
|
||
assumption
|
||
|
||
unfold circleIntegral 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 z R θ) = Complex.I • F (circleMap z R θ) := by
|
||
rw [← smul_assoc]
|
||
congr 1
|
||
simp
|
||
nth_rw 1 [mul_comm]
|
||
rw [← mul_assoc]
|
||
simp
|
||
apply inv_mul_cancel
|
||
apply circleMap_ne_center
|
||
exact Ne.symm (ne_of_lt hR)
|
||
have t'₁ {θ : ℝ} : circleMap 0 R θ = circleMap z R θ - z := by
|
||
exact Eq.symm (circleMap_sub_center z R θ)
|
||
simp_rw [← t'₁, t₁] at this
|
||
simp at this
|
||
|
||
have t₂ : Complex.reCLM (-Complex.I * (Complex.I * ∫ (x : ℝ) in (0)..2 * Real.pi, F (circleMap z R x))) = Complex.reCLM (-Complex.I * (2 * ↑Real.pi * Complex.I * F z)) := by
|
||
rw [this]
|
||
simp at t₂
|
||
have xx {x : ℝ} : (F (circleMap z R x)).re = f (circleMap z R x) := by
|
||
rw [← h₂F]
|
||
simp
|
||
simp
|
||
rw [Complex.dist_eq]
|
||
rw [circleMap_sub_center]
|
||
simp
|
||
rwa [abs_of_nonneg (le_of_lt hR)]
|
||
have x₁ {z : ℂ} : z.re = Complex.reCLM z := by rfl
|
||
rw [x₁] at t₂
|
||
rw [← ContinuousLinearMap.intervalIntegral_comp_comm] at t₂
|
||
simp at t₂
|
||
simp_rw [xx] at t₂
|
||
have x₁ {z : ℂ} : z.re = Complex.reCLM z := by rfl
|
||
rw [x₁] at t₂
|
||
have : Complex.reCLM (F z) = f z := by
|
||
apply h₂F
|
||
simp
|
||
exact gt_trans hρ hR
|
||
rw [this] at t₂
|
||
exact t₂
|
||
|
||
-- IntervalIntegrable (fun x => F (circleMap 0 1 x)) MeasureTheory.volume 0 (2 * Real.pi)
|
||
apply ContinuousOn.intervalIntegrable
|
||
apply ContinuousOn.comp reg₀F.continuousOn _
|
||
intro θ _
|
||
simp
|
||
rw [Complex.dist_eq, circleMap_sub_center]
|
||
simp
|
||
rwa [abs_of_nonneg (le_of_lt hR)]
|
||
apply Continuous.continuousOn
|
||
exact continuous_circleMap z R
|