299 lines
9.3 KiB
Plaintext
299 lines
9.3 KiB
Plaintext
import Mathlib.Analysis.Complex.CauchyIntegral
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import Mathlib.Analysis.Analytic.IsolatedZeros
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import Nevanlinna.analyticOn_zeroSet
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import Nevanlinna.harmonicAt_examples
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import Nevanlinna.harmonicAt_meanValue
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import Nevanlinna.specialFunctions_CircleIntegral_affine
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open Real
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/-
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noncomputable def Zeroset
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{f : ℂ → ℂ}
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{s : Set ℂ}
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(hf : ∀ z ∈ s, HolomorphicAt f z) :
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Set ℂ := by
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exact f⁻¹' {0} ∩ s
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noncomputable def ZeroFinset
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{f : ℂ → ℂ}
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(h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
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(h₂f : f 0 ≠ 0) :
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Finset ℂ := by
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let Z := f⁻¹' {0} ∩ Metric.closedBall (0 : ℂ) 1
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have hZ : Set.Finite Z := by
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dsimp [Z]
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rw [Set.inter_comm]
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apply finiteZeros
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-- Ball is preconnected
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apply IsConnected.isPreconnected
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apply Convex.isConnected
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exact convex_closedBall 0 1
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exact Set.nonempty_of_nonempty_subtype
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--
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exact isCompact_closedBall 0 1
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--
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intro x hx
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have A := (h₁f x hx)
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let B := HolomorphicAt_iff.1 A
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obtain ⟨s, h₁s, h₂s, h₃s⟩ := B
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apply DifferentiableOn.analyticAt (s := s)
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intro z hz
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apply DifferentiableAt.differentiableWithinAt
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apply h₃s
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exact hz
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exact IsOpen.mem_nhds h₁s h₂s
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--
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use 0
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constructor
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· simp
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· exact h₂f
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exact hZ.toFinset
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lemma ZeroFinset_mem_iff
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{f : ℂ → ℂ}
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(h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
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{h₂f : f 0 ≠ 0}
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(z : ℂ) :
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z ∈ ↑(ZeroFinset h₁f h₂f) ↔ z ∈ Metric.closedBall 0 1 ∧ f z = 0 := by
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dsimp [ZeroFinset]; simp
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tauto
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noncomputable def order
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{f : ℂ → ℂ}
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{h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z}
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{h₂f : f 0 ≠ 0} :
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ZeroFinset h₁f h₂f → ℕ := by
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intro i
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let A := ((ZeroFinset_mem_iff h₁f i).1 i.2).1
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let B := (h₁f i.1 A).analyticAt
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exact B.order.toNat
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-/
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theorem jensen_case_R_eq_one
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(f : ℂ → ℂ)
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(h₁f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt f z)
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(h'₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1))
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(h₂f : f 0 ≠ 0) :
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log ‖f 0‖ = -∑ᶠ s, (h'₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
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have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
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(convex_closedBall (0 : ℂ) 1).isPreconnected
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have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
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isCompact_closedBall 0 1
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have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
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use 0; simp; exact h₂f
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obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h'₁f h'₂f
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have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
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sorry
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let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order s).toNat * log ‖z - s‖
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have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
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intro z h₁z h₂z
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conv =>
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left
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arg 1
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rw [h₃F]
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rw [smul_eq_mul]
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rw [norm_mul]
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rw [norm_prod]
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left
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arg 2
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intro b
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rw [norm_pow]
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simp only [Complex.norm_eq_abs, Finset.univ_eq_attach]
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rw [Real.log_mul]
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rw [Real.log_prod]
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conv =>
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left
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left
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arg 2
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intro s
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rw [Real.log_pow]
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dsimp [G]
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abel
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-- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
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have : ∀ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0 := by
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intro s hs
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simp at hs
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simp
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intro h₂s
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rw [h₂s] at h₂z
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tauto
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exact this
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-- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
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have : ∀ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0 := by
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intro s hs
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simp at hs
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simp
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intro h₂s
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rw [h₂s] at h₂z
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tauto
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rw [Finset.prod_ne_zero_iff]
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exact this
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-- Complex.abs (F z) ≠ 0
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simp
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exact h₂F z h₁z
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have : ∫ (x : ℝ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ = ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) := by
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rw [intervalIntegral.integral_congr_ae]
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rw [MeasureTheory.ae_iff]
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apply Set.Countable.measure_zero
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simp
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have t₀ : {a | a ∈ Ι 0 (2 * π) ∧ ¬log ‖f (circleMap 0 1 a)‖ = G (circleMap 0 1 a)}
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⊆ (circleMap 0 1)⁻¹' (Metric.closedBall 0 1 ∩ f⁻¹' {0}) := by
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intro a ha
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simp at ha
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simp
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by_contra C
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have : (circleMap 0 1 a) ∈ Metric.closedBall 0 1 :=
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circleMap_mem_closedBall 0 (zero_le_one' ℝ) a
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exact ha.2 (decompose_f (circleMap 0 1 a) this C)
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apply Set.Countable.mono t₀
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apply Set.Countable.preimage_circleMap
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apply Set.Finite.countable
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let A := finiteZeros h₁U h₂U h'₁f h'₂f
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have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by
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ext z
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simp
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tauto
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rw [this]
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exact Set.Finite.image Subtype.val A
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exact Ne.symm (zero_ne_one' ℝ)
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have h₁Gi : ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, IntervalIntegrable (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π) := by
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-- This is hard. Need to invoke specialFunctions_CircleIntegral_affine.
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sorry
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have : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
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= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
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+ ∑ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
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dsimp [G]
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rw [intervalIntegral.integral_add]
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rw [intervalIntegral.integral_finset_sum]
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simp_rw [intervalIntegral.integral_const_mul]
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-- ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
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-- IntervalIntegrable (fun x => (h'₁f.order i).toNat *
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-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
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intro i hi
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apply IntervalIntegrable.const_mul
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--simp at this
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by_cases h₂i : ‖i.1‖ = 1
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-- case pos
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exact int'₂ h₂i
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-- case neg
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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by_contra ha'
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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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simp
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tauto
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp [h₂F]
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-- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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apply ContinuousAt.comp
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apply DifferentiableAt.continuousAt (𝕜 := ℂ )
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apply HolomorphicAt.differentiableAt
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simp [h'₁F]
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-- ContinuousAt (fun x => circleMap 0 1 x) x
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apply Continuous.continuousAt
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apply continuous_circleMap
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have : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
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= ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (fun x => (h'₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
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funext x
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simp
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rw [this]
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apply IntervalIntegrable.sum
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intro i h₂i
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apply IntervalIntegrable.const_mul
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--have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
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--simp at this
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by_cases h₂i : ‖i.1‖ = 1
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-- case pos
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exact int'₂ h₂i
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-- case neg
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--have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
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apply Continuous.intervalIntegrable
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apply continuous_iff_continuousAt.2
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intro x
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have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
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rfl
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rw [this]
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apply ContinuousAt.comp
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apply Real.continuousAt_log
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simp
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by_contra ha'
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conv at h₂i =>
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arg 1
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rw [← ha']
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rw [Complex.norm_eq_abs]
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rw [abs_circleMap_zero 1 x]
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simp
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tauto
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apply ContinuousAt.comp
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apply Complex.continuous_abs.continuousAt
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fun_prop
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have t₁ : (∫ (x : ℝ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
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let logAbsF := fun w ↦ Real.log ‖F w‖
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have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
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intro z hz
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apply logabs_of_holomorphicAt_is_harmonic
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apply h'₁F z hz
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exact h₂F z hz
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apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
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simp_rw [← Complex.norm_eq_abs] at this
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rw [t₁] at this
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sorry
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