working…

This commit is contained in:
Stefan Kebekus 2024-08-22 14:21:14 +02:00
parent 371b90c1c6
commit b818aa5c13
3 changed files with 99 additions and 20 deletions

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@ -33,6 +33,20 @@ theorem HolomorphicAt_iff
· assumption
theorem HolomorphicAt_analyticAt
{f : → F}
{x : } :
HolomorphicAt f x → AnalyticAt f x := by
intro hf
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
apply DifferentiableOn.analyticAt (s := s)
intro z hz
apply DifferentiableAt.differentiableWithinAt
apply h₃s
exact hz
exact IsOpen.mem_nhds h₁s h₂s
theorem HolomorphicAt_differentiableAt
{f : E → F}
{x : E} :

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@ -1,4 +1,5 @@
import Mathlib.Analysis.Complex.TaylorSeries
import Mathlib.Analysis.Complex.CauchyIntegral
--import Mathlib.Analysis.Complex.TaylorSeries
import Nevanlinna.cauchyRiemann
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace E]
@ -110,6 +111,21 @@ theorem HolomorphicAt_neg
exact h₂UF z hz
theorem HolomorphicAt.analyticAt
[CompleteSpace F]
{f : → F}
{x : } :
HolomorphicAt f x → AnalyticAt f x := by
intro hf
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HolomorphicAt_iff.1 hf
apply DifferentiableOn.analyticAt (s := s)
intro z hz
apply DifferentiableAt.differentiableWithinAt
apply h₃s
exact hz
exact IsOpen.mem_nhds h₁s h₂s
theorem HolomorphicAt.contDiffAt
[CompleteSpace F]
{f : → F}

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@ -8,28 +8,63 @@ import Nevanlinna.specialFunctions_CircleIntegral_affine
open Real
def ZeroFinset
noncomputable def ZeroFinset
{f : }
(h₁f : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z) :
(h₁f : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z)
(h₂f : f 0 ≠ 0) :
Finset := by
let Z := f⁻¹' {0} ∩ Metric.closedBall (0 : ) 1
have hZ : Set.Finite Z := by sorry
have hZ : Set.Finite Z := by
dsimp [Z]
rw [Set.inter_comm]
apply finiteZeros
-- Ball is preconnected
apply IsConnected.isPreconnected
apply Convex.isConnected
exact convex_closedBall 0 1
exact Set.nonempty_of_nonempty_subtype
--
exact isCompact_closedBall 0 1
--
intro x hx
have A := (h₁f x hx)
let B := HolomorphicAt_iff.1 A
obtain ⟨s, h₁s, h₂s, h₃s⟩ := B
apply DifferentiableOn.analyticAt (s := s)
intro z hz
apply DifferentiableAt.differentiableWithinAt
apply h₃s
exact hz
exact IsOpen.mem_nhds h₁s h₂s
--
use 0
constructor
· simp
· exact h₂f
exact hZ.toFinset
def order
{f : }
{hf : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z} :
ZeroFinset hf → := by sorry
lemma ZeroFinset_mem_iff
{f : }
(hf : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z)
(h₁f : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z)
{h₂f : f 0 ≠ 0}
(z : ) :
z ∈ ↑(ZeroFinset hf) ↔ z ∈ Metric.closedBall 0 1 ∧ f z = 0 := by
sorry
z ∈ ↑(ZeroFinset h₁f h₂f) ↔ z ∈ Metric.closedBall 0 1 ∧ f z = 0 := by
dsimp [ZeroFinset]; simp
tauto
noncomputable def order
{f : }
{h₁f : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z}
{h₂f : f 0 ≠ 0} :
ZeroFinset h₁f h₂f → := by
intro i
let A := ((ZeroFinset_mem_iff h₁f i).1 i.2).1
let B := (h₁f i.1 A).analyticAt
exact B.order.toNat
theorem jensen_case_R_eq_one
@ -37,14 +72,14 @@ theorem jensen_case_R_eq_one
(h₁f : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt f z)
(h'₁f : ∀ z ∈ Metric.closedBall (0 : ) 1, AnalyticAt f z)
(h₂f : f 0 ≠ 0) :
log ‖f 0‖ = -∑ s : ZeroFinset h₁f, order s * log (‖s.1‖⁻¹) + (2 * π )⁻¹ * ∫ (x : ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ := by
log ‖f 0‖ = -∑ s : (ZeroFinset h₁f h₂f), order s * log (‖s.1‖⁻¹) + (2 * π )⁻¹ * ∫ (x : ) in (0)..2 * π, log ‖f (circleMap 0 1 x)‖ := by
have F : := by sorry
have h₁F : ∀ z ∈ Metric.closedBall (0 : ) 1, HolomorphicAt F z := by sorry
have h₂F : ∀ z ∈ Metric.closedBall (0 : ) 1, F z ≠ 0 := by sorry
have h₃F : f = fun z ↦ (F z) * ∏ s : ZeroFinset h₁f, (z - s) ^ (order s) := by sorry
have h₃F : f = fun z ↦ (F z) * ∏ s : ZeroFinset h₁f h₂f, (z - s) ^ (order s) := by sorry
let G := fun z ↦ log ‖F z‖ + ∑ s : ZeroFinset h₁f, (order s) * log ‖z - s‖
let G := fun z ↦ log ‖F z‖ + ∑ s : ZeroFinset h₁f h₂f, (order s) * log ‖z - s‖
have decompose_f : ∀ z ∈ Metric.closedBall (0 : ) 1, f z ≠ 0 → log ‖f z‖ = G z := by
intro z h₁z h₂z
@ -123,11 +158,11 @@ theorem jensen_case_R_eq_one
exact ⟨Metric.mem_closedBall_self (zero_le_one' ), h₂f⟩
exact Ne.symm (zero_ne_one' )
have h₁Gi : ∀ i ∈ (ZeroFinset h₁f).attach, IntervalIntegrable (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π) := by
have h₁Gi : ∀ i ∈ (ZeroFinset h₁f h₂f).attach, 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.
sorry
have : ∫ (x : ) in (0)..2 * π, G (circleMap 0 1 x) = (∫ (x : ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x)))) + ∑ x ∈ (ZeroFinset h₁f).attach, ↑(order x) * ∫ (x_1 : ) in (0)..2 * π, log (Complex.abs (circleMap 0 1 x_1 - ↑x)) := by
have : ∫ (x : ) in (0)..2 * π, G (circleMap 0 1 x) = (∫ (x : ) in (0)..2 * π, log (Complex.abs (F (circleMap 0 1 x)))) + ∑ x ∈ (ZeroFinset h₁f h₂f).attach, ↑(order x) * ∫ (x_1 : ) 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]
@ -188,8 +223,8 @@ theorem jensen_case_R_eq_one
apply Continuous.continuousAt
apply continuous_circleMap
--
have : (fun x => ∑ s ∈ (ZeroFinset h₁f).attach, ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s)))
= ∑ s ∈ (ZeroFinset h₁f).attach, (fun x => ↑(order s) * log (Complex.abs (circleMap 0 1 x - ↑s))) := by
have : (fun x => ∑ s ∈ (ZeroFinset h₁f h₂f).attach, ↑(order s) * 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
funext x
simp
rw [this]
@ -225,5 +260,19 @@ theorem jensen_case_R_eq_one
apply Complex.continuous_abs.continuousAt
fun_prop
have t₁ : (∫ (x : ) in (0)..2 * Real.pi, log ‖F (circleMap 0 1 x)‖) = 2 * Real.pi * log ‖F 0‖ := by
let logAbsF := fun w ↦ Real.log ‖F w‖
have t₀ : ∀ z ∈ Metric.closedBall 0 1, HarmonicAt logAbsF z := by
intro z hz
apply logabs_of_holomorphicAt_is_harmonic
apply h₁F z hz
exact h₂F z hz
apply harmonic_meanValue₁ 1 Real.zero_lt_one t₀
simp_rw [← Complex.norm_eq_abs] at this
rw [t₁] at this
sorry