108 lines
3.4 KiB
Plaintext
108 lines
3.4 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
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import Nevanlinna.holomorphic_examples
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theorem harmonic_meanValue
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{f : ℂ → ℝ}
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{z : ℂ}
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(ρ R : ℝ)
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(hR : 0 < R)
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(hρ : R < ρ)
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(hf : ∀ x ∈ Metric.ball z ρ , HarmonicAt f x)
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:
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(∫ (x : ℝ) in (0)..2 * Real.pi, f (circleMap z R x)) = 2 * Real.pi * f z
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:= by
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obtain ⟨F, h₁F, h₂F⟩ := harmonic_is_realOfHolomorphic (gt_trans hρ hR) hf
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have hrρ : Metric.ball z R ⊆ Metric.ball z ρ := by
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intro x hx
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exact gt_trans hρ hx
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have reg₀F : DifferentiableOn ℂ F (Metric.ball z ρ) := by
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intro x hx
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apply DifferentiableAt.differentiableWithinAt
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apply HolomorphicAt.differentiableAt (h₁F x _)
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exact hx
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have reg₁F : DifferentiableOn ℂ F (Metric.ball z R) := by
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intro x hx
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apply DifferentiableAt.differentiableWithinAt
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apply HolomorphicAt.differentiableAt (h₁F x _)
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exact hrρ hx
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have : (∮ (x : ℂ) in C(z, R), (x - z)⁻¹ • F x) = (2 * ↑Real.pi * Complex.I) • F z := by
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let s : Set ℂ := ∅
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let hs : s.Countable := Set.countable_empty
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let _ : ℂ := 0
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have hw : (z : ℂ) ∈ Metric.ball z R := Metric.mem_ball_self hR
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have hc : ContinuousOn F (Metric.closedBall z R) := by
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apply reg₀F.continuousOn.mono
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intro x hx
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simp at hx
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simp
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linarith
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have hd : ∀ x ∈ Metric.ball z R \ s, DifferentiableAt ℂ F x := by
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intro x hx
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let A := reg₁F x hx.1
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apply A.differentiableAt
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apply (IsOpen.mem_nhds_iff ?hs).mpr
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exact hx.1
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exact Metric.isOpen_ball
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let CIF := Complex.circleIntegral_sub_inv_smul_of_differentiable_on_off_countable hs hw hc hd
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simp at CIF
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assumption
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unfold circleIntegral at this
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simp_rw [deriv_circleMap] at this
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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
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rw [← smul_assoc]
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congr 1
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simp
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nth_rw 1 [mul_comm]
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rw [← mul_assoc]
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simp
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apply inv_mul_cancel
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apply circleMap_ne_center
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exact Ne.symm (ne_of_lt hR)
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have t'₁ {θ : ℝ} : circleMap 0 R θ = circleMap z R θ - z := by
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exact Eq.symm (circleMap_sub_center z R θ)
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simp_rw [← t'₁, t₁] at this
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simp at this
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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
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rw [this]
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simp at t₂
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have xx {x : ℝ} : (F (circleMap z R x)).re = f (circleMap z R x) := by
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rw [← h₂F]
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simp
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simp
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rw [Complex.dist_eq]
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rw [circleMap_sub_center]
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simp
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rwa [abs_of_nonneg (le_of_lt hR)]
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have x₁ {z : ℂ} : z.re = Complex.reCLM z := by rfl
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rw [x₁] at t₂
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rw [← ContinuousLinearMap.intervalIntegral_comp_comm] at t₂
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simp at t₂
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simp_rw [xx] at t₂
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have x₁ {z : ℂ} : z.re = Complex.reCLM z := by rfl
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rw [x₁] at t₂
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have : Complex.reCLM (F z) = f z := by
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apply h₂F
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simp
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exact gt_trans hρ hR
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rw [this] at t₂
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exact t₂
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-- IntervalIntegrable (fun x => F (circleMap 0 1 x)) MeasureTheory.volume 0 (2 * Real.pi)
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apply ContinuousOn.intervalIntegrable
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apply ContinuousOn.comp reg₀F.continuousOn _
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intro θ _
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simp
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rw [Complex.dist_eq, circleMap_sub_center]
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simp
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rwa [abs_of_nonneg (le_of_lt hR)]
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apply Continuous.continuousOn
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exact continuous_circleMap z R
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