working

Nevanlinna/specialFunctions_CircleIntegral_affine.lean

@@ -36,10 +36,13 @@ lemma l₂ {x : ℝ} : ‖(circleMap 0 1 x) - a‖ = ‖1 - (circleMap 0 1 (-x))
     dsimp [circleMap]
     simp
 
+
+-- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ < 1.
+
 lemma int₀
   {a : ℂ}
   (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
   · -- case: a = 0
@@ -108,6 +111,8 @@ lemma int₀
   exact A
 
 
+-- Integral of log ‖circleMap 0 1 x - 1‖
+
 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
@@ -141,34 +146,43 @@ lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
   apply IntervalIntegrable.const_mul
   exact intervalIntegrable_log_sin
 
+lemma logAffineHelper {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
+  dsimp [Complex.abs]
+  rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
+  congr
+  calc Complex.normSq (circleMap 0 1 x - 1)
+  _ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
+    dsimp [circleMap]
+    rw [Complex.normSq_apply]
+    simp
+  _ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
+    ring
+  _ = 2 - 2 * cos x := by
+    rw [sin_sq_add_cos_sq]
+    norm_num
+  _ = 2 - 2 * cos (2 * (x / 2)) := by
+    rw [← mul_div_assoc]
+    congr; norm_num
+  _ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
+    rw [cos_two_mul]
+    ring
+  _ = 4 * sin (x / 2) ^ 2 := by
+    nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
+    ring
+
+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
 
-  have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
-    dsimp [Complex.abs]
-    rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
-    congr
-    calc Complex.normSq (circleMap 0 1 x - 1)
-    _ = (cos x - 1) * (cos x - 1) + sin x * sin x := by
-      dsimp [circleMap]
-      rw [Complex.normSq_apply]
-      simp
-    _ = sin x ^ 2 + cos x ^ 2 + 1 - 2 * cos x := by
-      ring
-    _ = 2 - 2 * cos x := by
-      rw [sin_sq_add_cos_sq]
-      norm_num
-    _ = 2 - 2 * cos (2 * (x / 2)) := by
-      rw [← mul_div_assoc]
-      congr; norm_num
-    _ = 4 - 4 * Real.cos (x / 2) ^ 2 := by
-      rw [cos_two_mul]
-      ring
-    _ = 4 * sin (x / 2) ^ 2 := by
-      nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
-      ring
-  simp_rw [this]
+  simp_rw [logAffineHelper]
   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

Nevanlinna/specialFunctions_Integral_log_sin.lean

@@ -254,6 +254,19 @@ theorem intervalIntegral.integral_congr_volume
   _ = 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
   rw [← intervalIntegral.integral_add_adjacent_intervals (a := 0) (b := π / 2) (c := π)]