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This commit is contained in:
Stefan Kebekus
2024-08-22 13:09:03 +02:00
parent db9ce54bcf
commit 371b90c1c6
3 changed files with 272 additions and 8 deletions
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82
Nevanlinna/holomorphic_JensenFormula2.lean
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@@ -3,6 +3,7 @@ import Mathlib.Analysis.Analytic.IsolatedZeros
import Nevanlinna.analyticOn_zeroSet
import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
import Nevanlinna.specialFunctions_CircleIntegral_affine
open Real
@@ -88,6 +89,7 @@ theorem jensen_case_R_eq_one
tauto
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
rw [intervalIntegral.integral_congr_ae]
rw [MeasureTheory.ae_iff]
apply Set.Countable.measure_zero
@@ -133,11 +135,16 @@ theorem jensen_case_R_eq_one
-- ∀ i ∈ (ZeroFinset h₁f).attach, IntervalIntegrable (fun x => ↑(order i) *
-- log (Complex.abs (circleMap 0 1 x - ↑i))) MeasureTheory.volume 0 (2 * π)
-- This won't work, because the function **is not** continuous. Need to fix.
intro i hi
apply IntervalIntegrable.const_mul
exact h₁Gi i hi
have : i.1 ∈ Metric.closedBall (0 : ℂ) 1 := by exact ((ZeroFinset_mem_iff h₁f i).1 i.2).1
simp at this
by_cases h₂i : ‖i.1‖ = 1
-- case pos
exact int'₂ h₂i
-- case neg
have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2
@@ -150,14 +157,73 @@ theorem jensen_case_R_eq_one
simp
by_contra ha'
let A := ha i
rw [← ha'] at A
simp at A
conv at h₂i =>
arg 1
rw [← ha']
rw [Complex.norm_eq_abs]
rw [abs_circleMap_zero 1 x]
simp
tauto
apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt
fun_prop
-- IntervalIntegrable (fun x => log (Complex.abs (F (circleMap 0 1 x)))) MeasureTheory.volume 0 (2 * π)
apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2
intro x
have : (fun x => log (Complex.abs (F (circleMap 0 1 x)))) = log ∘ Complex.abs ∘ F ∘ (fun x ↦ circleMap 0 1 x) :=
rfl
rw [this]
apply ContinuousAt.comp
apply Real.continuousAt_log
simp [h₂F]
--
apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt
apply ContinuousAt.comp
apply DifferentiableAt.continuousAt (𝕜 := ℂ )
apply HolomorphicAt.differentiableAt
simp [h₁F]
--
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
funext x
simp
rw [this]
apply IntervalIntegrable.sum
intro i h₂i
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
by_cases h₂i : ‖i.1‖ = 1
-- case pos
exact int'₂ h₂i
-- case neg
have : i.1 ∈ Metric.ball (0 : ℂ) 1 := by sorry
apply Continuous.intervalIntegrable
apply continuous_iff_continuousAt.2
intro x
have : (fun x => log (Complex.abs (circleMap 0 1 x - ↑i))) = log ∘ Complex.abs ∘ (fun x ↦ circleMap 0 1 x - ↑i) :=
rfl
rw [this]
apply ContinuousAt.comp
apply Real.continuousAt_log
simp
by_contra ha'
conv at h₂i =>
arg 1
rw [← ha']
rw [Complex.norm_eq_abs]
rw [abs_circleMap_zero 1 x]
simp
tauto
apply ContinuousAt.comp
apply Complex.continuous_abs.continuousAt
fun_prop
sorry
sorry

109
Nevanlinna/periodic_integrability.lean Normal file
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@@ -0,0 +1,109 @@
import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.MeasureTheory.Integral.CircleIntegral
import Nevanlinna.specialFunctions_Integral_log_sin
import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
import Mathlib.Algebra.Periodic
open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral
theorem periodic_integrability₁
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E}
{t T : ℝ}
{n : ℕ}
(h₁f : Function.Periodic f T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume t (t + n * T) := by
induction' n with n hn
simp
apply IntervalIntegrable.trans (b := t + n * T)
exact hn
let A := IntervalIntegrable.comp_sub_right h₂f (n * T)
have : f = fun x ↦ f (x - n * T) := by simp [Function.Periodic.sub_nat_mul_eq h₁f n]
simp_rw [← this] at A
have : (t + T + ↑n * T) = (t + ↑(n + 1) * T) := by simp; ring
simp_rw [this] at A
exact A
theorem periodic_integrability₂
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E}
{t T : ℝ}
{n : ℕ}
(h₁f : Function.Periodic f T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume (t - n * T) t := by
induction' n with n hn
simp
--
apply IntervalIntegrable.trans (b := t - n * T)
--
have A := IntervalIntegrable.comp_add_right h₂f (((n + 1): ℕ) * T)
have : f = fun x ↦ f (x + ((n + 1) : ℕ) * T) := by
funext x
have : x = x + ↑(n + 1) * T - ↑(n + 1) * T := by ring
nth_rw 1 [this]
rw [Function.Periodic.sub_nat_mul_eq h₁f]
simp_rw [← this] at A
have : (t + T - ↑(n + 1) * T) = (t - ↑n * T) := by simp; ring
simp_rw [this] at A
exact A
--
exact hn
theorem periodic_integrability₃
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E}
{t T : ℝ}
{n₁ n₂ : ℕ}
(h₁f : Function.Periodic f T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume (t - n₁ * T) (t + n₂ * T) := by
apply IntervalIntegrable.trans (b := t)
exact periodic_integrability₂ h₁f h₂f
exact periodic_integrability₁ h₁f h₂f
theorem periodic_integrability4
{E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E}
{t T : ℝ}
{a₁ a₂ : ℝ}
(h₁f : Function.Periodic f T)
(hT : 0 < T)
(h₂f : IntervalIntegrable f MeasureTheory.volume t (t + T)) :
IntervalIntegrable f MeasureTheory.volume a₁ a₂ := by
obtain ⟨n₁, hn₁⟩ : ∃ n₁ : ℕ, t - n₁ * T ≤ min a₁ a₂ := by
obtain ⟨n₁, hn₁⟩ := exists_nat_ge ((t -min a₁ a₂) / T)
use n₁
rw [sub_le_iff_le_add]
rw [div_le_iff hT] at hn₁
rw [sub_le_iff_le_add] at hn₁
rw [add_comm]
exact hn₁
obtain ⟨n₂, hn₂⟩ : ∃ n₂ : ℕ, max a₁ a₂ ≤ t + n₂ * T := by
obtain ⟨n₂, hn₂⟩ := exists_nat_ge ((max a₁ a₂ - t) / T)
use n₂
rw [← sub_le_iff_le_add]
rw [div_le_iff hT] at hn₂
linarith
have : Set.uIcc a₁ a₂ ⊆ Set.uIcc (t - n₁ * T) (t + n₂ * T) := by
apply Set.uIcc_subset_uIcc_iff_le.mpr
constructor
· calc min (t - ↑n₁ * T) (t + ↑n₂ * T)
_ ≤ (t - ↑n₁ * T) := by exact min_le_left (t - ↑n₁ * T) (t + ↑n₂ * T)
_ ≤ min a₁ a₂ := by exact hn₁
· calc max a₁ a₂
_ ≤ t + n₂ * T := by exact hn₂
_ ≤ max (t - ↑n₁ * T) (t + ↑n₂ * T) := by exact le_max_right (t - ↑n₁ * T) (t + ↑n₂ * T)
apply IntervalIntegrable.mono_set _ this
exact periodic_integrability₃ h₁f h₂f

89
Nevanlinna/specialFunctions_CircleIntegral_affine.lean
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@@ -3,12 +3,19 @@ import Mathlib.MeasureTheory.Integral.CircleIntegral
import Nevanlinna.specialFunctions_Integral_log_sin
import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
import Nevanlinna.periodic_integrability
open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral
-- Integrability of periodic functions
-- Lemmas for the circleMap
lemma l₀ {x₁ x₂ : ℝ} : (circleMap 0 1 x₁) * (circleMap 0 1 x₂) = circleMap 0 1 (x₁+x₂) := by
dsimp [circleMap]
simp
@@ -187,6 +194,23 @@ lemma int'₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖
apply IntervalIntegrable.const_mul
exact intervalIntegrable_log_sin
lemma int''₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖ for arbitrary intervals
∀ (t₁ t₂ : ℝ), IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - 1‖) volume t₁ t₂ := by
intro t₁ t₂
apply periodic_integrability4 (T := 2 * π) (t := 0)
--
have : (fun x => log ‖circleMap 0 1 x - 1‖) = (fun x => log ‖x - 1‖) ∘ (circleMap 0 1) := rfl
rw [this]
apply Function.Periodic.comp
exact periodic_circleMap 0 1
--
exact two_pi_pos
--
rw [zero_add]
exact int'₁
lemma int₁ :
∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
@@ -213,6 +237,71 @@ lemma int₁ :
exact int₁₁
-- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ = 1
lemma int'₂
{a : ℂ}
(ha : ‖a‖ = 1) :
IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - a‖) volume 0 (2 * π) := by
simp_rw [l₂]
have {x : ℝ} : log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv =>
arg 1
intro x
rw [this]
rw [IntervalIntegrable.iff_comp_neg]
let f₁ := fun w ↦ log ‖1 - circleMap 0 1 w * a‖
have {x : ℝ} : log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * π) := by
dsimp [f₁]
congr 4
let A := periodic_circleMap 0 1 x
simp at A
exact id (Eq.symm A)
conv =>
arg 1
intro x
arg 0
intro w
rw [this]
simp
have : 0 = 2 * π - 2 * π := by ring
rw [this]
have : -(2 * π ) = 0 - 2 * π := by ring
rw [this]
apply IntervalIntegrable.comp_add_right _ (2 * π) --f₁ (2 * π)
dsimp [f₁]
have : ∃ ζ, a = circleMap 0 1 ζ := by
apply Set.exists_range_iff.mp
use a
simp
exact ha
obtain ⟨ζ, hζ⟩ := this
rw [hζ]
simp_rw [l₀]
have : 2 * π = (2 * π + ζ) - ζ := by ring
rw [this]
have : 0 = ζ - ζ := by ring
rw [this]
have : (fun w => log (Complex.abs (1 - circleMap 0 1 (w + ζ)))) = fun x ↦ (fun w ↦ log (Complex.abs (1 - circleMap 0 1 (w)))) (x + ζ) := rfl
rw [this]
apply IntervalIntegrable.comp_add_right (f := (fun w ↦ log (Complex.abs (1 - circleMap 0 1 (w))))) _ ζ
have : Function.Periodic (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) (2 * π) := by
have : (fun x ↦ log (Complex.abs (1 - circleMap 0 1 x))) = ( (fun x ↦ log (Complex.abs (1 - x))) ∘ (circleMap 0 1) ) := by rfl
rw [this]
apply Function.Periodic.comp
exact periodic_circleMap 0 1
let A := int''₁ (2 * π + ζ) ζ
have {x : ℝ} : ‖circleMap 0 1 x - 1‖ = Complex.abs (1 - circleMap 0 1 x) := AbsoluteValue.map_sub Complex.abs (circleMap 0 1 x) 1
simp_rw [this] at A
exact A
lemma int₂
{a : ℂ}
(ha : ‖a‖ = 1) :