diff --git a/Nevanlinna/specialFunctions_CircleIntegral_affine.lean b/Nevanlinna/specialFunctions_CircleIntegral_affine.lean
index 03603be..bfe992a 100644
--- a/Nevanlinna/specialFunctions_CircleIntegral_affine.lean
+++ b/Nevanlinna/specialFunctions_CircleIntegral_affine.lean
@@ -130,6 +130,7 @@ lemma int₀
 
 -- Integral of log ‖circleMap 0 1 x - 1‖
 
+-- integral
 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
@@ -248,6 +249,7 @@ lemma int₁ :
 
 -- Integral of log ‖circleMap 0 1 x - a‖, for ‖a‖ = 1
 
+-- integrability
 lemma int'₂
   {a : ℂ}
   (ha : ‖a‖ = 1) :
@@ -310,7 +312,7 @@ lemma int'₂
   simp_rw [this] at A
   exact A
 
-
+-- integral
 lemma int₂
   {a : ℂ}
   (ha : ‖a‖ = 1) :
@@ -370,7 +372,7 @@ lemma int₂
   simp_rw [this]
   exact int₁
 
-
+-- integral
 lemma int₃
   {a : ℂ}
   (ha : a ∈ Metric.closedBall 0 1) :
@@ -383,7 +385,7 @@ lemma int₃
     simp
     linarith
 
-
+-- integral
 lemma int₄
   {a : ℂ}
   {R : ℝ}
@@ -485,3 +487,33 @@ lemma int₄
         exact Ne.symm (ne_of_lt hR)
       rw [div_self this] at h₂a
       tauto
+
+
+lemma intervalIntegrable_logAbs_circleMap_sub_const
+  {a c : ℂ}
+  {r : ℝ}
+  (hr : r ≠ 0) :
+  IntervalIntegrable (fun x ↦ log ‖circleMap c r x - a‖) volume 0 (2 * π) := by
+
+  have {x : ℝ} : log ‖circleMap c r x - a‖ = log ‖r * (circleMap 0 1 x - r⁻¹ * (a - c))‖ := by
+    unfold circleMap
+    congr 2
+    simp
+    rw [mul_sub]
+    rw [← mul_assoc]
+    simp [hr]
+    ring
+  simp_rw [this]
+
+  have {x : ℝ} : log ‖r * (circleMap 0 1 x - r⁻¹ * (a - c))‖ = log r + log ‖(circleMap 0 1 x - r⁻¹ * (a - c))‖ := by
+    rw [norm_mul]
+    rw [log_mul]
+    simp
+    --
+    simp [hr]
+    --
+
+
+
+    sorry
+  sorry