Update specialFunctions_Integrals.lean

Nevanlinna/specialFunctions_Integrals.lean

@@ -5,6 +5,8 @@ import Mathlib.MeasureTheory.Integral.CircleIntegral
 open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
+-- The following theorem was suggested by Gareth Ma on Zulip
+
 theorem logInt
   {t : ℝ}
   (ht : 0 < t) :
@@ -63,6 +65,50 @@ theorem logInt
 
 lemma int₁ :
   ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
+
+
+
+  have {x : ℝ} : Complex.normSq (circleMap 0 1 x - 1) = 2 - 2 * cos x := by
+    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
+
+
+  have {x : ℝ} : 2 - 2 * cos x = 4 * sin (x / 2) ^ 2 := by
+    calc 2 - 2 * cos x
+    _ = 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]
+      norm_num
+    _ = 4 * sin (x / 2) ^ 2 := by
+      nth_rw 1 [← mul_one 4]
+      nth_rw 1 [← sin_sq_add_cos_sq (x / 2)]
+      rw [mul_add]
+      abel
+
+
+
+
+
+
+  dsimp [Complex.abs]
+  sorry
+
+
+
+
+    sorry
+
+
+
 
   sorry