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Nevanlinna/holomorphic_JensenFormula.lean

@@ -100,6 +100,13 @@ lemma int₀
   exact A
 
 
+lemma int₁ :
+  ∫ (x : ℝ) in (0)..2 * Real.pi, Real.log ‖circleMap 0 1 x - 1‖ = 0 := by
+  dsimp [circleMap]
+
+  sorry
+
+
 theorem jensen_case_R_eq_one
   (f : ℂ → ℂ)
   (h₁f : Differentiable ℂ f)

Nevanlinna/specialFunctions.lean

@@ -0,0 +1,54 @@
+import Mathlib.Analysis.SpecialFunctions.Integrals
+
+theorem intervalIntegral.intervalIntegrable_log'
+  {a : ℝ}
+  {b : ℝ}
+  {μ : MeasureTheory.Measure ℝ}
+  [MeasureTheory.IsLocallyFiniteMeasure μ]
+  (ha : 0 < a) :
+  IntervalIntegrable Real.log μ 0 a
+  := by
+
+  sorry
+
+theorem integral_log₀
+  {b : ℝ}
+  (hb : 0 < b) :
+  ∫ (x : ℝ) in (0)..b, Real.log x = b * (Real.log b - 1) := by
+  apply?
+  exact integral_log h
+
+
+open Real Nat Set Finset
+
+open scoped Real Interval
+
+--variable {a b : ℝ} (n : ℕ)
+
+namespace intervalIntegral
+
+--open MeasureTheory
+
+--variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ)
+
+#check integral_mul_deriv_eq_deriv_mul
+
+theorem integral_log₁
+  (h : (0 : ℝ) ∉ [[a, b]]) :
+  ∫ x in a..b, log x = b * log b - a * log a - b + a := by
+
+  have h' : ∀ x ∈ [[a, b]], x ≠ 0 :=
+    fun x (hx : x ∈ [[a, b]]) => ne_of_mem_of_not_mem hx h
+  have heq :  ∀ x ∈ [[a, b]], x * x⁻¹ = 1 :=
+    fun x hx => mul_inv_cancel (h' x hx)
+
+  let A := fun x hx => hasDerivAt_log (h' x hx)
+
+  convert integral_mul_deriv_eq_deriv_mul A (fun x _ => hasDerivAt_id x)
+
+  convert integral_mul_deriv_eq_deriv_mul A
+    (fun x _ => hasDerivAt_id x) (continuousOn_inv₀.mono <|
+      subset_compl_singleton_iff.mpr h).intervalIntegrable
+        continuousOn_const.intervalIntegrable using 1 <;>
+
+  simp [integral_congr heq, mul_comm, ← sub_add]