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This commit is contained in:
Stefan Kebekus 2024-08-15 12:10:18 +02:00
parent 10f88298c0
commit 23dfdd3716
2 changed files with 38 additions and 4 deletions
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33
Nevanlinna/specialFunctions_CircleIntegral_affine.lean
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@ -2,6 +2,7 @@ import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.MeasureTheory.Integral.CircleIntegral import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Measure.Restrict import Mathlib.MeasureTheory.Measure.Restrict
import Nevanlinna.specialFunctions_Integral_log_sin
open scoped Interval Topology open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral open Real Filter MeasureTheory intervalIntegral
@ -10,7 +11,37 @@ open Real Filter MeasureTheory intervalIntegral
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
sorry
have t₁ {x : ℝ} : x ∈ Set.Ioo 0 π → log (4 * sin x ^ 2) = log 4 + 2 * log (sin x) := by
intro hx
rw [log_mul, log_pow]
rfl
exact Ne.symm (NeZero.ne' 4)
apply pow_ne_zero 2
apply (fun a => Ne.symm (ne_of_lt a))
exact sin_pos_of_mem_Ioo hx
have t₂ : Set.EqOn (fun y ↦ log (4 * sin y ^ 2)) (fun y ↦ log 4 + 2 * log (sin y)) (Set.Ioo 0 π) := by
intro x hx
simp
rw [t₁ hx]
rw [intervalIntegral.integral_congr_volume pi_pos t₂]
rw [intervalIntegral.integral_add]
rw [intervalIntegral.integral_const_mul]
simp
rw [integral_log_sin₂]
have : (4 : ℝ) = 2 * 2 := by norm_num
rw [this, log_mul]
ring
norm_num
norm_num
-- IntervalIntegrable (fun x => log 4) volume 0 π
simp
-- IntervalIntegrable (fun x => 2 * log (sin x)) volume 0 π
apply IntervalIntegrable.const_mul
exact intervalIntegrable_log_sin
lemma int₁ : lemma int₁ :

9
Nevanlinna/specialFunctions_Integral_log_sin.lean
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@ -1,7 +1,5 @@
import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.MeasureTheory.Integral.CircleIntegral
import Mathlib.MeasureTheory.Measure.Restrict
open scoped Interval Topology open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral open Real Filter MeasureTheory intervalIntegral
@ -226,6 +224,7 @@ theorem intervalIntegrable_log_cos : IntervalIntegrable (log ∘ cos) volume 0 (
apply IntervalIntegrable.symm apply IntervalIntegrable.symm
rwa [← this] rwa [← this]
theorem intervalIntegral.integral_congr_volume theorem intervalIntegral.integral_congr_volume
{E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E] {E : Type u_3} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E} {f : ℝ → E}
@ -255,7 +254,7 @@ theorem intervalIntegral.integral_congr_volume
_ = 0 := volume_singleton _ = 0 := volume_singleton
lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = -log 2 * π/2 := by lemma integral_log_sin₁ : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = -log 2 * π/2 := by
have t₁ {x : ℝ} : x ∈ Set.Ioo 0 (π / 2) → log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by have t₁ {x : ℝ} : x ∈ Set.Ioo 0 (π / 2) → log (sin (2 * x)) = log 2 + log (sin x) + log (cos x) := by
intro hx intro hx
@ -352,3 +351,7 @@ lemma integral_log_sin : ∫ (x : ℝ) in (0)..(π / 2), log (sin x) = -log 2 *
exact intervalIntegrable_log_cos exact intervalIntegrable_log_cos
-- --
linarith [pi_pos] linarith [pi_pos]
lemma integral_log_sin₂ : ∫ (x : ℝ) in (0)..π, log (sin x) = -log 2 * π := by
sorry