66 lines
2.2 KiB
Plaintext
66 lines
2.2 KiB
Plaintext
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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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(hf : ∀ z, HarmonicAt f z)
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(R : ℝ)
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(hR : R > 0) :
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(∫ (x : ℝ) in (0)..2 * Real.pi, f (circleMap 0 R x)) = 2 * Real.pi * f 0
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:= by
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obtain ⟨F, h₁F, h₂F⟩ := harmonic_is_realOfHolomorphic hf
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have regF : Differentiable ℂ F := fun z ↦ HolomorphicAt.differentiableAt (h₁F z)
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have : (∮ (z : ℂ) in C(0, R), z⁻¹ • F z) = (2 * ↑Real.pi * Complex.I) • F 0 := 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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let hw : (0 : ℂ) ∈ Metric.ball 0 R := Metric.mem_ball_self hR
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let hc : ContinuousOn F (Metric.closedBall 0 R) := by
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apply Continuous.continuousOn
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exact regF.continuous
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let hd : ∀ x ∈ Metric.ball 0 R \ s, DifferentiableAt ℂ F x := by
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intro x _
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exact regF x
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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 0 R θ) = Complex.I • F (circleMap 0 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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simp_rw [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 0 R x))) = Complex.reCLM (-Complex.I * (2 * ↑Real.pi * Complex.I * F 0)) := by
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rw [this]
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simp at t₂
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have xx {z : ℂ} : (F z).re = f z := by
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rw [← h₂F]
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simp
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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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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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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 Continuous.intervalIntegrable
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apply Continuous.comp
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exact regF.continuous
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exact continuous_circleMap 0 R
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