Update holomorphic_JensenFormula.lean

Nevanlinna/holomorphic_JensenFormula.lean

@@ -33,6 +33,12 @@ lemma int₀
   (ha : a ∈ Metric.ball 0 1) :
   ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - a‖ = 0 := by
 
+  by_cases h₁a : a = 0
+  · -- case: a = 0
+    rw [h₁a]
+    simp
+
+  -- case: a ≠ 0
   simp_rw [l₂]
   have {x : ℝ} : Real.log ‖1 - circleMap 0 1 (-x) * a‖ = (fun w ↦ Real.log ‖1 - circleMap 0 1 (w) * a‖) (-x) := by rfl
   conv =>
@@ -58,12 +64,37 @@ lemma int₀
   simp
   dsimp [f₁]
 
+  let ρ := ‖a‖⁻¹
+  have hρ : 1 < ρ := by
+    apply one_lt_inv_iff.mpr
+    constructor
+    · exact norm_pos_iff'.mpr h₁a
+    · exact mem_ball_zero_iff.mp ha
+
   let F := fun z ↦ Real.log ‖1 - z * a‖
 
-  have hf : ∀ x ∈ Metric.ball 0 2 , HarmonicAt F x := by
-    sorry
+  have hf : ∀ x ∈ Metric.ball 0 ρ, HarmonicAt F x := by
+    intro x hx
+    apply logabs_of_holomorphicAt_is_harmonic
+    apply Differentiable.holomorphicAt
+    fun_prop
+    apply sub_ne_zero_of_ne
+    by_contra h
+    have : ‖x * a‖ < 1 := by
+      calc ‖x * a‖
+      _ = ‖x‖ * ‖a‖ := by exact NormedField.norm_mul' x a
+      _ < ρ * ‖a‖ := by
+        apply (mul_lt_mul_right _).2
+        exact mem_ball_zero_iff.mp hx
+        exact norm_pos_iff'.mpr h₁a
+      _ = 1 := by
+        dsimp [ρ]
+        apply inv_mul_cancel
+        exact (AbsoluteValue.ne_zero_iff Complex.abs).mpr h₁a
+    rw [← h] at this
+    simp at this
 
-  let A := harmonic_meanValue 2 1 Real.zero_lt_one one_lt_two hf
+  let A := harmonic_meanValue ρ 1 Real.zero_lt_one hρ hf
   dsimp [F] at A
   simp at A
   exact A