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This commit is contained in:
Stefan Kebekus 2024-08-15 16:00:25 +02:00
parent 23dfdd3716
commit 44dc57ed39
7 changed files with 237 additions and 137 deletions

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@ -47,6 +47,24 @@ theorem HarmonicAt_iff
· apply (IsOpen.mem_nhds_iff h₁s).2 h₂s · apply (IsOpen.mem_nhds_iff h₁s).2 h₂s
· exact h₂f · exact h₂f
theorem HarmonicAt_isOpen
(f : → F) :
IsOpen { x : | HarmonicAt f x } := by
rw [← subset_interior_iff_isOpen]
intro x hx
simp at hx
obtain ⟨s, h₁s, h₂s, h₃s⟩ := HarmonicAt_iff.1 hx
use s
constructor
· simp
constructor
· exact h₁s
· intro x hx
simp
rw [HarmonicAt_iff]
use s
· exact h₂s
theorem HarmonicAt_eventuallyEq {f₁ f₂ : → F} {x : } (h : f₁ =ᶠ[nhds x] f₂) : HarmonicAt f₁ x ↔ HarmonicAt f₂ x := by theorem HarmonicAt_eventuallyEq {f₁ f₂ : → F} {x : } (h : f₁ =ᶠ[nhds x] f₂) : HarmonicAt f₁ x ↔ HarmonicAt f₂ x := by
constructor constructor

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@ -1,5 +1,5 @@
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Nevanlinna.complexHarmonic import Nevanlinna.harmonicAt
import Nevanlinna.holomorphicAt import Nevanlinna.holomorphicAt
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace F] [CompleteSpace F] variable {F : Type*} [NormedAddCommGroup F] [NormedSpace F] [CompleteSpace F]

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@ -1,3 +1,5 @@
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Analysis.Complex.CauchyIntegral import Mathlib.Analysis.Complex.CauchyIntegral
import Nevanlinna.holomorphic_examples import Nevanlinna.holomorphic_examples
@ -104,3 +106,26 @@ theorem harmonic_meanValue
rwa [abs_of_nonneg (le_of_lt hR)] rwa [abs_of_nonneg (le_of_lt hR)]
apply Continuous.continuousOn apply Continuous.continuousOn
exact continuous_circleMap z R exact continuous_circleMap z R
/- Probably not optimal yet. We want a statements that requires harmonicity in
the interior and continuity on the closed ball.
-/
theorem harmonic_meanValue₁
{f : }
{z : }
(R : )
(hR : 0 < R)
(hf : ∀ x ∈ Metric.closedBall z R , HarmonicAt f x) :
(∫ (x : ) in (0)..2 * Real.pi, f (circleMap z R x)) = 2 * Real.pi * f z := by
obtain ⟨e, h₁e, h₂e⟩ := IsCompact.exists_thickening_subset_open (isCompact_closedBall z R) (HarmonicAt_isOpen f) hf
rw [thickening_closedBall] at h₂e
apply harmonic_meanValue (e + R) R
assumption
norm_num
assumption
exact fun x a => h₂e a
assumption
linarith

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@ -1,110 +1,6 @@
import Nevanlinna.harmonicAt_examples import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue import Nevanlinna.harmonicAt_meanValue
import Nevanlinna.specialFunctions_CircleIntegral_affine
lemma l₀ {x₁ x₂ : } : (circleMap 0 1 x₁) * (circleMap 0 1 x₂) = circleMap 0 1 (x₁+x₂) := by
dsimp [circleMap]
simp
rw [add_mul, Complex.exp_add]
lemma l₁ {x : } : ‖circleMap 0 1 x‖ = 1 := by
rw [Complex.norm_eq_abs, abs_circleMap_zero]
simp
lemma l₂ {x : } : ‖(circleMap 0 1 x) - a‖ = ‖1 - (circleMap 0 1 (-x)) * a‖ := by
calc ‖(circleMap 0 1 x) - a‖
_ = 1 * ‖(circleMap 0 1 x) - a‖ := by
exact Eq.symm (one_mul ‖circleMap 0 1 x - a‖)
_ = ‖(circleMap 0 1 (-x))‖ * ‖(circleMap 0 1 x) - a‖ := by
rw [l₁]
_ = ‖(circleMap 0 1 (-x)) * ((circleMap 0 1 x) - a)‖ := by
exact Eq.symm (NormedField.norm_mul' (circleMap 0 1 (-x)) (circleMap 0 1 x - a))
_ = ‖(circleMap 0 1 (-x)) * (circleMap 0 1 x) - (circleMap 0 1 (-x)) * a‖ := by
rw [mul_sub]
_ = ‖(circleMap 0 1 0) - (circleMap 0 1 (-x)) * a‖ := by
rw [l₀]
simp
_ = ‖1 - (circleMap 0 1 (-x)) * a‖ := by
congr
dsimp [circleMap]
simp
lemma int₀
{a : }
(ha : a ∈ Metric.ball 0 1) :
∫ (x : ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by
by_cases h₁a : a = 0
· -- case: a = 0
rw [h₁a]
simp
-- case: a ≠ 0
simp_rw [l₂]
have {x : } : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv =>
left
arg 1
intro x
rw [this]
rw [intervalIntegral.integral_comp_neg ((fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖))]
let f₁ := fun w ↦ Real.log ‖1 - circleMap 0 1 w * a‖
have {x : } : Real.log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * Real.pi) := by
dsimp [f₁]
congr 4
let A := periodic_circleMap 0 1 x
simp at A
exact id (Eq.symm A)
conv =>
left
arg 1
intro x
rw [this]
rw [intervalIntegral.integral_comp_add_right f₁ (2 * Real.pi)]
simp
dsimp [f₁]
let ρ := ‖a‖⁻¹
have hρ : 1 < ρ := by
apply one_lt_inv_iff.mpr
constructor
· exact norm_pos_iff'.mpr h₁a
· exact mem_ball_zero_iff.mp ha
let F := fun z ↦ Real.log ‖1 - z * a‖
have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by
intro x hx
apply logabs_of_holomorphicAt_is_harmonic
apply Differentiable.holomorphicAt
fun_prop
apply sub_ne_zero_of_ne
by_contra h
have : ‖x * a‖ < 1 := by
calc ‖x * a‖
_ = ‖x‖ * ‖a‖ := by exact NormedField.norm_mul' x a
_ < ρ * ‖a‖ := by
apply (mul_lt_mul_right _).2
exact mem_ball_zero_iff.mp hx
exact norm_pos_iff'.mpr h₁a
_ = 1 := by
dsimp [ρ]
apply inv_mul_cancel
exact (AbsoluteValue.ne_zero_iff Complex.abs).mpr h₁a
rw [← h] at this
simp at this
let A := harmonic_meanValue ρ 1 Real.zero_lt_one hρ hf
dsimp [F] at A
simp at A
exact A
lemma int₁ :
∫ (x : ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - 1‖ = 0 := by
dsimp [circleMap]
sorry
theorem jensen_case_R_eq_one theorem jensen_case_R_eq_one

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@ -1,4 +1,4 @@
import Nevanlinna.complexHarmonic import Nevanlinna.harmonicAt
import Nevanlinna.holomorphicAt import Nevanlinna.holomorphicAt
import Nevanlinna.holomorphic_primitive import Nevanlinna.holomorphic_primitive
import Nevanlinna.mathlibAddOn import Nevanlinna.mathlibAddOn

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@ -1,14 +1,112 @@
import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.MeasureTheory.Integral.Periodic
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.MeasureTheory.Integral.CircleIntegral import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Measure.Restrict
import Nevanlinna.specialFunctions_Integral_log_sin import Nevanlinna.specialFunctions_Integral_log_sin
import Nevanlinna.harmonicAt_examples
import Nevanlinna.harmonicAt_meanValue
open scoped Interval Topology open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral open Real Filter MeasureTheory intervalIntegral
lemma l₀ {x₁ x₂ : } : (circleMap 0 1 x₁) * (circleMap 0 1 x₂) = circleMap 0 1 (x₁+x₂) := by
dsimp [circleMap]
simp
rw [add_mul, Complex.exp_add]
lemma l₁ {x : } : ‖circleMap 0 1 x‖ = 1 := by
rw [Complex.norm_eq_abs, abs_circleMap_zero]
simp
lemma l₂ {x : } : ‖(circleMap 0 1 x) - a‖ = ‖1 - (circleMap 0 1 (-x)) * a‖ := by
calc ‖(circleMap 0 1 x) - a‖
_ = 1 * ‖(circleMap 0 1 x) - a‖ := by
exact Eq.symm (one_mul ‖circleMap 0 1 x - a‖)
_ = ‖(circleMap 0 1 (-x))‖ * ‖(circleMap 0 1 x) - a‖ := by
rw [l₁]
_ = ‖(circleMap 0 1 (-x)) * ((circleMap 0 1 x) - a)‖ := by
exact Eq.symm (NormedField.norm_mul' (circleMap 0 1 (-x)) (circleMap 0 1 x - a))
_ = ‖(circleMap 0 1 (-x)) * (circleMap 0 1 x) - (circleMap 0 1 (-x)) * a‖ := by
rw [mul_sub]
_ = ‖(circleMap 0 1 0) - (circleMap 0 1 (-x)) * a‖ := by
rw [l₀]
simp
_ = ‖1 - (circleMap 0 1 (-x)) * a‖ := by
congr
dsimp [circleMap]
simp
lemma int₀
{a : }
(ha : a ∈ Metric.ball 0 1) :
∫ (x : ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by
by_cases h₁a : a = 0
· -- case: a = 0
rw [h₁a]
simp
-- case: a ≠ 0
simp_rw [l₂]
have {x : } : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv =>
left
arg 1
intro x
rw [this]
rw [intervalIntegral.integral_comp_neg ((fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖))]
let f₁ := fun w ↦ Real.log ‖1 - circleMap 0 1 w * a‖
have {x : } : Real.log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * Real.pi) := by
dsimp [f₁]
congr 4
let A := periodic_circleMap 0 1 x
simp at A
exact id (Eq.symm A)
conv =>
left
arg 1
intro x
rw [this]
rw [intervalIntegral.integral_comp_add_right f₁ (2 * Real.pi)]
simp
dsimp [f₁]
let ρ := ‖a‖⁻¹
have hρ : 1 < ρ := by
apply one_lt_inv_iff.mpr
constructor
· exact norm_pos_iff'.mpr h₁a
· exact mem_ball_zero_iff.mp ha
let F := fun z ↦ Real.log ‖1 - z * a‖
have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by
intro x hx
apply logabs_of_holomorphicAt_is_harmonic
apply Differentiable.holomorphicAt
fun_prop
apply sub_ne_zero_of_ne
by_contra h
have : ‖x * a‖ < 1 := by
calc ‖x * a‖
_ = ‖x‖ * ‖a‖ := by exact NormedField.norm_mul' x a
_ < ρ * ‖a‖ := by
apply (mul_lt_mul_right _).2
exact mem_ball_zero_iff.mp hx
exact norm_pos_iff'.mpr h₁a
_ = 1 := by
dsimp [ρ]
apply inv_mul_cancel
exact (AbsoluteValue.ne_zero_iff Complex.abs).mpr h₁a
rw [← h] at this
simp at this
let A := harmonic_meanValue ρ 1 Real.zero_lt_one hρ hf
dsimp [F] at A
simp at A
exact A
lemma int₁₁ : ∫ (x : ) in (0)..π, log (4 * sin x ^ 2) = 0 := by lemma int₁₁ : ∫ (x : ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
@ -91,3 +189,63 @@ lemma int₁ :
rw [this] rw [this]
simp simp
exact int₁₁ exact int₁₁
lemma int₂
{a : }
(ha : ‖a‖ = 1) :
∫ x in (0)..(2 * Real.pi), log ‖circleMap 0 1 x - a‖ = 0 := by
simp_rw [l₂]
have {x : } : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
conv =>
left
arg 1
intro x
rw [this]
rw [intervalIntegral.integral_comp_neg ((fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖))]
let f₁ := fun w ↦ Real.log ‖1 - circleMap 0 1 w * a‖
have {x : } : Real.log ‖1 - circleMap 0 1 x * a‖ = f₁ (x + 2 * Real.pi) := by
dsimp [f₁]
congr 4
let A := periodic_circleMap 0 1 x
simp at A
exact id (Eq.symm A)
conv =>
left
arg 1
intro x
rw [this]
rw [intervalIntegral.integral_comp_add_right f₁ (2 * Real.pi)]
simp
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]
rw [intervalIntegral.integral_comp_add_right (f := fun x ↦ log (Complex.abs (1 - circleMap 0 1 (x))))]
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 := Function.Periodic.intervalIntegral_add_eq this ζ 0
simp at A
simp
rw [add_comm]
rw [A]
have {x : } : log (Complex.abs (1 - circleMap 0 1 x)) = log ‖circleMap 0 1 x - 1‖ := by
rw [← AbsoluteValue.map_neg Complex.abs]
simp
simp_rw [this]
exact int₁

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@ -254,6 +254,31 @@ theorem intervalIntegral.integral_congr_volume
_ = 0 := volume_singleton _ = 0 := volume_singleton
lemma integral_log_sin₀ : ∫ (x : ) in (0)..π, log (sin x) = 2 * ∫ (x : ) in (0)..(π / 2), log (sin x) := by
rw [← intervalIntegral.integral_add_adjacent_intervals (a := 0) (b := π / 2) (c := π)]
conv =>
left
right
arg 1
intro x
rw [← sin_pi_sub]
rw [intervalIntegral.integral_comp_sub_left (fun x ↦ log (sin x)) π]
have : π - π / 2 = π / 2 := by linarith
rw [this]
simp
ring
-- IntervalIntegrable (fun x => log (sin x)) volume 0 (π / 2)
exact intervalIntegrable_log_sin₂
-- IntervalIntegrable (fun x => log (sin x)) volume (π / 2) π
apply intervalIntegrable_log_sin.mono_set
rw [Set.uIcc_of_le, Set.uIcc_of_le]
apply Set.Icc_subset_Icc_left
linarith [pi_pos]
linarith [pi_pos]
linarith [pi_pos]
lemma integral_log_sin₁ : ∫ (x : ) in (0)..(π / 2), log (sin x) = -log 2 * π/2 := by lemma integral_log_sin₁ : ∫ (x : ) in (0)..(π / 2), log (sin x) = -log 2 * π/2 := by
have t₁ {x : } : x ∈ Set.Ioo 0 (π / 2) → log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by have t₁ {x : } : x ∈ Set.Ioo 0 (π / 2) → log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by
@ -295,30 +320,7 @@ lemma integral_log_sin₁ : ∫ (x : ) in (0)..(π / 2), log (sin x) = -log
simp simp
have : 2 * (π / 2) = π := by linarith have : 2 * (π / 2) = π := by linarith
rw [this] rw [this]
rw [integral_log_sin₀]
have : ∫ (x : ) in (0)..π, log (sin x) = 2 * ∫ (x : ) in (0)..(π / 2), log (sin x) := by
rw [← intervalIntegral.integral_add_adjacent_intervals (a := 0) (b := π / 2) (c := π)]
conv =>
left
right
arg 1
intro x
rw [← sin_pi_sub]
rw [intervalIntegral.integral_comp_sub_left (fun x ↦ log (sin x)) π]
have : π - π / 2 = π / 2 := by linarith
rw [this]
simp
ring
-- IntervalIntegrable (fun x => log (sin x)) volume 0 (π / 2)
exact intervalIntegrable_log_sin₂
-- IntervalIntegrable (fun x => log (sin x)) volume (π / 2) π
apply intervalIntegrable_log_sin.mono_set
rw [Set.uIcc_of_le, Set.uIcc_of_le]
apply Set.Icc_subset_Icc_left
linarith [pi_pos]
linarith [pi_pos]
linarith [pi_pos]
rw [this]
have : ∫ (x : ) in (0)..(π / 2), log (sin x) = ∫ (x : ) in (0)..(π / 2), log (cos x) := by have : ∫ (x : ) in (0)..(π / 2), log (sin x) = ∫ (x : ) in (0)..(π / 2), log (cos x) := by
conv => conv =>
@ -354,4 +356,5 @@ lemma integral_log_sin₁ : ∫ (x : ) in (0)..(π / 2), log (sin x) = -log
lemma integral_log_sin₂ : ∫ (x : ) in (0)..π, log (sin x) = -log 2 * π := by lemma integral_log_sin₂ : ∫ (x : ) in (0)..π, log (sin x) = -log 2 * π := by
sorry rw [integral_log_sin₀, integral_log_sin₁]
ring