Update holomorphic_JensenFormula2.lean
This commit is contained in:
Stefan Kebekus
parent
47e1bfe35e
commit
b91e3677c0
|
|
@ -7,30 +7,31 @@ import Nevanlinna.specialFunctions_CircleIntegral_affine
|
|||
|
||||
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
|
||||
(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) :
|
||||
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
|
||||
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
|
||||
intro z h₁z
|
||||
apply AnalyticAt.holomorphicAt
|
||||
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
|
||||
intro z h₁z h₂z
|
||||
|
|
@ -59,7 +60,7 @@ theorem jensen_case_R_eq_one
|
|||
abel
|
||||
|
||||
-- ∀ 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
|
||||
simp at hs
|
||||
simp
|
||||
|
|
@ -69,7 +70,7 @@ theorem jensen_case_R_eq_one
|
|||
exact this
|
||||
|
||||
-- ∏ 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
|
||||
simp at hs
|
||||
simp
|
||||
|
|
@ -104,7 +105,7 @@ theorem jensen_case_R_eq_one
|
|||
apply Set.Countable.mono t₀
|
||||
apply Set.Countable.preimage_circleMap
|
||||
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
|
||||
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)
|
||||
= (∫ (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]
|
||||
rw [intervalIntegral.integral_add]
|
||||
rw [intervalIntegral.integral_finset_sum]
|
||||
|
|
@ -175,8 +176,8 @@ theorem jensen_case_R_eq_one
|
|||
apply Continuous.continuousAt
|
||||
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)))
|
||||
= ∑ 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
|
||||
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 h₁f h'₂f).toFinset, (fun x => (h₁f.order s).toNat * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
|
||||
funext x
|
||||
simp
|
||||
rw [this]
|
||||
|
|
@ -252,7 +253,7 @@ theorem jensen_case_R_eq_one
|
|||
rw [log_prod]
|
||||
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
|
||||
intro x ⟨h₁x, _⟩
|
||||
|
|
@ -260,7 +261,7 @@ theorem jensen_case_R_eq_one
|
|||
|
||||
dsimp [AnalyticOn.order] 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
|
||||
|
|
|
|||
Loading…
Reference in New Issue