From bc8fed96b0c33aeabe508e77f28cec5ff558f2a3 Mon Sep 17 00:00:00 2001
From: Stefan Kebekus <kebekus@users.noreply.github.com>
Date: Tue, 13 Aug 2024 09:46:30 +0200
Subject: [PATCH] Update specialFunctions_Integrals.lean

---
 Nevanlinna/specialFunctions_Integrals.lean | 48 +++++++++++++++++++++-
 1 file changed, 47 insertions(+), 1 deletion(-)

diff --git a/Nevanlinna/specialFunctions_Integrals.lean b/Nevanlinna/specialFunctions_Integrals.lean
index 13dad97..0ceb7da 100644
--- a/Nevanlinna/specialFunctions_Integrals.lean
+++ b/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
-  dsimp [circleMap]
+
+
+
+  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