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Update holomorphic_JensenFormula2.lean

This commit is contained in:
Stefan Kebekus 2024-09-11 09:07:49 +02:00
parent e901f241cc
commit 3bead7a9bf
1 changed files with 20 additions and 24 deletions
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42
Nevanlinna/holomorphic_JensenFormula2.lean
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@ -7,7 +7,7 @@ import Nevanlinna.specialFunctions_CircleIntegral_affine
open Real open Real
/-
noncomputable def Zeroset noncomputable def Zeroset
{f : ℂ → ℂ} {f : ℂ → ℂ}
{s : Set ℂ} {s : Set ℂ}
@ -72,7 +72,7 @@ noncomputable def order
let A := ((ZeroFinset_mem_iff h₁f i).1 i.2).1 let A := ((ZeroFinset_mem_iff h₁f i).1 i.2).1
let B := (h₁f i.1 A).analyticAt let B := (h₁f i.1 A).analyticAt
exact B.order.toNat exact B.order.toNat
-/
theorem jensen_case_R_eq_one theorem jensen_case_R_eq_one
(f : ℂ → ℂ) (f : ℂ → ℂ)
@ -180,29 +180,28 @@ theorem jensen_case_R_eq_one
exact Ne.symm (zero_ne_one' ℝ) exact Ne.symm (zero_ne_one' ℝ)
have h₁Gi : ∀ i ∈ (ZeroFinset h₁f h₂f).attach, IntervalIntegrable (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π) := by 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
-- This is hard. Need to invoke specialFunctions_CircleIntegral_affine. -- This is hard. Need to invoke specialFunctions_CircleIntegral_affine.
sorry sorry
have : ∫ (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 ∈ (ZeroFinset h₁f h₂f).attach, ↑(order x) * ∫ (x_1 : ℝ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by have : ∫ (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
dsimp [G] dsimp [G]
rw [intervalIntegral.integral_add] rw [intervalIntegral.integral_add]
rw [intervalIntegral.integral_finset_sum] rw [intervalIntegral.integral_finset_sum]
simp_rw [intervalIntegral.integral_const_mul] simp_rw [intervalIntegral.integral_const_mul]
-- ∀ i ∈ (ZeroFinset h₁f).attach, IntervalIntegrable (fun x => ↑(order i) * -- ∀ i ∈ (finiteZeros h₁U h₂U h'₁f h'₂f).toFinset,
-- IntervalIntegrable (fun x => (h'₁f.order i).toNat *
-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π) -- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
intro i hi intro i hi
apply IntervalIntegrable.const_mul apply IntervalIntegrable.const_mul
have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1 --simp at this
simp at this
by_cases h₂i : ‖i.1‖ = 1 by_cases h₂i : ‖i.1‖ = 1
-- case pos -- case pos
exact int'₂ h₂i exact int'₂ h₂i
-- case neg -- case neg
have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
apply Continuous.intervalIntegrable apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2 apply continuous_iff_continuousAt.2
intro x intro x
@ -234,32 +233,32 @@ theorem jensen_case_R_eq_one
apply ContinuousAt.comp apply ContinuousAt.comp
apply Real.continuousAt_log apply Real.continuousAt_log
simp [h₂F] simp [h₂F]
-- -- ContinuousAt (⇑Complex.abs ∘ F ∘ fun x => circleMap 0 1 x) x
apply ContinuousAt.comp apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt apply Complex.continuous_abs.continuousAt
apply ContinuousAt.comp apply ContinuousAt.comp
apply DifferentiableAt.continuousAt (𝕜 := ℂ ) apply DifferentiableAt.continuousAt (𝕜 := ℂ )
apply HolomorphicAt.differentiableAt apply HolomorphicAt.differentiableAt
simp [h₁F] simp [h'₁F]
-- -- ContinuousAt (fun x => circleMap 0 1 x) x
apply Continuous.continuousAt apply Continuous.continuousAt
apply continuous_circleMap apply continuous_circleMap
--
have : (fun x => ∑ s ∈ (ZeroFinset h₁f h₂f).attach, ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s))) 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 ∈ (ZeroFinset h₁f h₂f).attach, (fun x => ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s))) := by = ∑ 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
funext x funext x
simp simp
rw [this] rw [this]
apply IntervalIntegrable.sum apply IntervalIntegrable.sum
intro i h₂i intro i h₂i
apply IntervalIntegrable.const_mul apply IntervalIntegrable.const_mul
have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1 --have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := i.2
simp at this --simp at this
by_cases h₂i : ‖i.1‖ = 1 by_cases h₂i : ‖i.1‖ = 1
-- case pos -- case pos
exact int'₂ h₂i exact int'₂ h₂i
-- case neg -- case neg
have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry --have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
apply Continuous.intervalIntegrable apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2 apply continuous_iff_continuousAt.2
intro x intro x
@ -287,16 +286,13 @@ theorem jensen_case_R_eq_one
have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
intro z hz intro z hz
apply logabs_of_holomorphicAt_is_harmonic apply logabs_of_holomorphicAt_is_harmonic
apply h₁F z hz apply h'₁F z hz
exact h₂F z hz exact h₂F z hz
apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀ apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
simp_rw [← Complex.norm_eq_abs] at this simp_rw [← Complex.norm_eq_abs] at this
rw [t₁] at this rw [t₁] at this
--let Z₁ := (ZeroFinset h₁f h₂f) ∩ (Metric.ball 0 1)
let Z₂ := { x : ZeroFinset h₁f h₂f | ‖x.1‖ = 1 }
sorry sorry