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Update holomorphic_JensenFormula2.lean

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Stefan Kebekus 2024-09-11 11:09:03 +02:00
parent 47e1bfe35e
commit b91e3677c0
1 changed files with 18 additions and 17 deletions
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Nevanlinna/holomorphic_JensenFormula2.lean
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@ -7,30 +7,31 @@ import Nevanlinna.specialFunctions_CircleIntegral_affine
open Real open Real
lemma h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
(convex_closedBall (0 : ℂ) 1).isPreconnected
lemma h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
isCompact_closedBall 0 1
theorem jensen_case_R_eq_one theorem jensen_case_R_eq_one
(f : ℂ → ℂ) (f : ℂ → ℂ)
(h'₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1)) (h₁f : AnalyticOn ℂ f (Metric.closedBall (0 : ℂ) 1))
(h₂f : f 0 ≠ 0) : (h₂f : f 0 ≠ 0) :
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 log ‖f 0‖ = -∑ᶠ s, (h₁f.order s).toNat * log (‖s.1‖⁻¹) + (2 * π)⁻¹ * ∫ (x : ℝ) in (0)..(2 * π), log ‖f (circleMap 0 1 x)‖ := by
have h₁U : IsPreconnected (Metric.closedBall (0 : ℂ) 1) :=
(convex_closedBall (0 : ℂ) 1).isPreconnected
have h₂U : IsCompact (Metric.closedBall (0 : ℂ) 1) :=
isCompact_closedBall 0 1
have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by have h'₂f : ∃ u ∈ (Metric.closedBall (0 : ℂ) 1), f u ≠ 0 := by
use 0; simp; exact h₂f use 0; simp; exact h₂f
obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h'₁f h'₂f obtain ⟨F, h₁F, h₂F, h₃F⟩ := AnalyticOnCompact.eliminateZeros₂ h₁U h₂U h₁f h'₂f
have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by have h'₁F : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, HolomorphicAt F z := by
intro z h₁z intro z h₁z
apply AnalyticAt.holomorphicAt apply AnalyticAt.holomorphicAt
exact h₁F z h₁z exact h₁F z h₁z
let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, (h'₁f.order s).toNat * log ‖z - s‖ let G := fun z ↦ log ‖F z‖ + ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log ‖z - s‖
have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by have decompose_f : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
intro z h₁z h₂z intro z h₁z h₂z
@ -59,7 +60,7 @@ theorem jensen_case_R_eq_one
abel abel
-- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0 -- ∀ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
have : ∀ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0 := by have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
intro s hs intro s hs
simp at hs simp at hs
simp simp
@ -69,7 +70,7 @@ theorem jensen_case_R_eq_one
exact this exact this
-- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0 -- ∏ x ∈ ⋯.toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0
have : ∀ x ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h'₁f.order x).toNat ≠ 0 := by have : ∀ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, Complex.abs (z - ↑x) ^ (h₁f.order x).toNat ≠ 0 := by
intro s hs intro s hs
simp at hs simp at hs
simp simp
@ -104,7 +105,7 @@ theorem jensen_case_R_eq_one
apply Set.Countable.mono t₀ apply Set.Countable.mono t₀
apply Set.Countable.preimage_circleMap apply Set.Countable.preimage_circleMap
apply Set.Finite.countable apply Set.Finite.countable
let A := finiteZeros h₁U h₂U h'₁f h'₂f let A := finiteZeros h₁U h₂U h₁f h'₂f
have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by have : (Metric.closedBall 0 1 ∩ f ⁻¹' {0}) = (Metric.closedBall 0 1).restrict f ⁻¹' {0} := by
ext z ext z
@ -117,7 +118,7 @@ theorem jensen_case_R_eq_one
have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x) have decompose_int_G : ∫ (x : ℝ) in (0)..2 * π, G (circleMap 0 1 x)
= (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x)))) = (∫ (x : ℝ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x))))
+ ∑ 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 + ∑ x ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order x).toNat * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
dsimp [G] dsimp [G]
rw [intervalIntegral.integral_add] rw [intervalIntegral.integral_add]
rw [intervalIntegral.integral_finset_sum] rw [intervalIntegral.integral_finset_sum]
@ -175,8 +176,8 @@ theorem jensen_case_R_eq_one
apply Continuous.continuousAt apply Continuous.continuousAt
apply continuous_circleMap apply continuous_circleMap
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))) have : (fun x => ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s)))
= ∑ 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 = ∑ s ∈ (finiteZeros h₁U h₂U h₁f h'₂f).toFinset, (fun x => (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
funext x funext x
simp simp
rw [this] rw [this]
@ -252,7 +253,7 @@ theorem jensen_case_R_eq_one
rw [log_prod] rw [log_prod]
simp simp
rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset)] rw [finsum_eq_sum_of_support_subset _ (s := (finiteZeros h₁U h₂U h₁f h'₂f).toFinset)]
simp simp
simp simp
intro x ⟨h₁x, _⟩ intro x ⟨h₁x, _⟩
@ -260,7 +261,7 @@ theorem jensen_case_R_eq_one
dsimp [AnalyticOn.order] at h₁x dsimp [AnalyticOn.order] at h₁x
simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x simp only [Function.mem_support, ne_eq, Nat.cast_eq_zero, not_or] at h₁x
exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h'₁f x) h₁x exact AnalyticAt.supp_order_toNat (AnalyticOn.order.proof_1 h₁f x) h₁x
-- --
intro x hx intro x hx