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nevanlinna/test.lean

@@ -2,6 +2,7 @@ import Mathlib.Analysis.Complex.CauchyIntegral
 
 #check DiffContOnCl.circleIntegral_sub_inv_smul
 
+open Real
 
 theorem CauchyIntegralFormula :
   ∀
@@ -10,8 +11,45 @@ theorem CauchyIntegralFormula :
   {f : ℂ → ℂ}, -- Holomorphic function
   DiffContOnCl ℂ f (Metric.ball 0 R)
   → w ∈ Metric.ball 0 R
-  → (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * Real.pi * Complex.I) • f w := by
+  → (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * π * Complex.I) • f w := by
 
   exact DiffContOnCl.circleIntegral_sub_inv_smul
 
 #check CauchyIntegralFormula
+#check HasDerivAt.continuousAt
+#check Real.log
+#check Complex.log
+#check Complex.exp
+
+theorem SimpleCauchyFormula :
+  ∀
+  {R : ℝ} -- Radius of the ball
+  {w : ℂ} -- Point in the interior of the ball
+  {f : ℂ → ℂ}, -- Holomorphic function
+  Differentiable ℂ f
+  → w ∈ Metric.ball 0 R
+  → (∮ (z : ℂ) in C(0, R), (z - w)⁻¹ • f z) = (2 * Real.pi * Complex.I) • f w := by
+
+  intro R w f fHyp
+
+  apply DiffContOnCl.circleIntegral_sub_inv_smul
+
+  constructor
+  · exact Differentiable.differentiableOn fHyp
+  · suffices Continuous f from by
+      exact Continuous.continuousOn this
+    rw [continuous_iff_continuousAt]
+    intro x
+    exact DifferentiableAt.continuousAt (fHyp x)
+
+
+theorem JensenFormula₂ :
+  ∀
+  {R : ℝ} -- Radius of the ball
+  {w : ℂ} -- Point in the interior of the ball
+  {f : ℂ → ℂ}, -- Holomorphic function
+  Differentiable ℂ f
+  → ∀ z ∈ Metric.ball 0 R, f z ≠ 0
+  → (∮ (z : ℂ) in C(0, R), Complex.log ‖f z‖ ) = 2 * π * R * log ‖f 0‖ := by
+
+  sorry