kebekus
/
nevanlinna
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

working

This commit is contained in:
Stefan Kebekus 2024-08-22 08:19:01 +02:00
parent 567b08aa5b
commit 77dea4115e
2 changed files with 52 additions and 25 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

28
Nevanlinna/specialFunctions_CircleIntegral_affine.lean
View File

@ -36,10 +36,13 @@ lemma l₂ {x : ℝ} : ‖(circleMap 0 1 x) - a‖ = ‖1 - (circleMap 0 1 (-x))
dsimp [circleMap] dsimp [circleMap]
simp simp
-- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ < 1.
lemma int₀ lemma int₀
{a : ℂ} {a : ℂ}
(ha : a ∈ Metric.ball 0 1) : (ha : a ∈ Metric.ball 0 1) :
∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by ∫ (x : ℝ) in (0)..2 * π, log ‖circleMap 0 1 x - a‖ = 0 := by
by_cases h₁a : a = 0 by_cases h₁a : a = 0
· -- case: a = 0 · -- case: a = 0
@ -108,6 +111,8 @@ lemma int₀
exact A exact A
-- Integral of log ‖circleMap 0 1 x - 1‖
lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
have t₁ {x : ℝ} : x ∈ Set.Ioo 0 π → log (4 * sin x ^ 2) = log 4 + 2 * log (sin x) := by have t₁ {x : ℝ} : x ∈ Set.Ioo 0 π → log (4 * sin x ^ 2) = log 4 + 2 * log (sin x) := by
@ -141,11 +146,7 @@ lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
apply IntervalIntegrable.const_mul apply IntervalIntegrable.const_mul
exact intervalIntegrable_log_sin exact intervalIntegrable_log_sin
lemma logAffineHelper {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
lemma int₁ :
∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
dsimp [Complex.abs] dsimp [Complex.abs]
rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))] rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
congr congr
@ -168,7 +169,20 @@ lemma int₁ :
_ = 4 * sin (x / 2) ^ 2 := by _ = 4 * sin (x / 2) ^ 2 := by
nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)] nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
ring ring
simp_rw [this]
lemma int'₁ : -- Integrability of log ‖circleMap 0 1 x - 1‖
IntervalIntegrable (fun x ↦ log ‖circleMap 0 1 x - 1‖) volume 0 (2 * π) := by
simp_rw [logAffineHelper]
rw [← IntervalIntegrable.comp_mul_left_iff (c := 2) (Ne.symm (NeZero.ne' 2))]
simp
sorry
lemma int₁ :
∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
simp_rw [logAffineHelper]
simp simp
have : ∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2) = 2 * ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) := by have : ∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2) = 2 * ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) := by

13
Nevanlinna/specialFunctions_Integral_log_sin.lean
View File

@ -254,6 +254,19 @@ theorem intervalIntegral.integral_congr_volume
_ = 0 := volume_singleton _ = 0 := volume_singleton
theorem IntervalIntegrable.integral_congr_Ioo
{E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f g : ℝ → E}
{a b : ℝ}
(hab : a ≤ b)
(hfg : Set.EqOn f g (Set.Ioo a b)) :
IntervalIntegrable f volume a b ↔ IntervalIntegrable g volume a b := by
rw [intervalIntegrable_iff_integrableOn_Ioo_of_le hab]
rw [MeasureTheory.integrableOn_congr_fun hfg measurableSet_Ioo]
rw [← intervalIntegrable_iff_integrableOn_Ioo_of_le hab]
lemma integral_log_sin₀ : ∫ (x : ℝ) in (0)..π, log (sin x) = 2 * ∫ (x : ℝ) in (0)..(π / 2), log (sin x) := by lemma integral_log_sin₀ : ∫ (x : ℝ) in (0)..π, log (sin x) = 2 * ∫ (x : ℝ) in (0)..(π / 2), log (sin x) := by
rw [← intervalIntegral.integral_add_adjacent_intervals (a := 0) (b := π / 2) (c := π)] rw [← intervalIntegral.integral_add_adjacent_intervals (a := 0) (b := π / 2) (c := π)]