Nevanlinna/harmonicAt_meanValue.lean

@@ -7,7 +7,8 @@ theorem harmonic_meanValue
   (ρ R : ℝ)
   (hR : 0 < R)
   (hρ : R < ρ)
-  (hf : ∀ x ∈ Metric.ball z ρ , HarmonicAt f x) :
+  (hf : ∀ x ∈ Metric.ball z ρ , HarmonicAt f x)
+  :
   (∫ (x : ℝ) in (0)..2 * Real.pi, f (circleMap z R x)) = 2 * Real.pi * f z
   := by
 

Nevanlinna/holomorphic_JensenFormula2.lean

@@ -18,7 +18,7 @@ lemma xx
       let A := hf z hz
       let B := A.order
 
-      exact (A.order : ⊤)
+      exact A.order
     else
       exact 0
 

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]

Nevanlinna/specialFunctions_Integrals.lean

@@ -1,162 +0,0 @@
-import Mathlib.Analysis.SpecialFunctions.Integrals
-import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
-import Mathlib.MeasureTheory.Integral.CircleIntegral
-import Mathlib.MeasureTheory.Measure.Restrict
-
-open scoped Interval Topology
-open Real Filter MeasureTheory intervalIntegral
-
--- The following theorem was suggested by Gareth Ma on Zulip
-
-example : IntervalIntegrable (log ∘ sin) volume 0 1 := by
-
-  have int_log : IntervalIntegrable log volume 0 1 := by sorry
-
-  apply IntervalIntegrable.mono_fun' (g := log)
-
-  exact int_log
-
-  -- AEStronglyMeasurable (log ∘ sin) (volume.restrict (Ι 0 1))
-  apply ContinuousOn.aestronglyMeasurable
-  apply ContinuousOn.comp (t := Ι 0 1)
-  apply ContinuousOn.mono (s := {0}ᶜ)
-  exact continuousOn_log
-  intro x hx
-  by_contra contra
-  simp at contra
-  rw [contra] at hx
-  rw [Set.left_mem_uIoc] at hx
-  linarith
-  exact continuousOn_sin
-  --
-  rw [Set.uIoc_of_le (zero_le_one' ℝ)]
-  exact fun x hx ↦ ⟨sin_pos_of_pos_of_le_one hx.1 hx.2, sin_le_one x⟩
-  --
-  exact measurableSet_uIoc
-  --
-
-  have : ∀ x ∈ (Ι 0 1), ‖(log ∘ sin) x‖ ≤ log x := by sorry
-  dsimp [EventuallyLE]
-  rw [MeasureTheory.ae_restrict_iff]
-  apply MeasureTheory.ae_of_all
-  exact this
-
-  --intro x
-  rw [MeasureTheory.ae_iff]
-  simp
-
-  rw [MeasureTheory.ae_iff]
-  simp
-
-
-  sorry
-
-
-theorem logInt
-  {t : ℝ}
-  (ht : 0 < t) :
-  ∫ x in (0 : ℝ)..t, log x = t * log t - t := by
-  rw [← integral_add_adjacent_intervals (b := 1)]
-  trans (-1) + (t * log t - t + 1)
-  · congr
-    · -- ∫ x in 0..1, log x = -1, same proof as before
-      rw [integral_eq_sub_of_hasDerivAt_of_tendsto (f := fun x ↦ x * log x - x) (fa := 0) (fb := -1)]
-      · simp
-      · simp
-      · intro x hx
-        norm_num at hx
-        convert (hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x) using 1
-        norm_num
-      · rw [← neg_neg log]
-        apply IntervalIntegrable.neg
-        apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
-        · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
-        · intro x hx
-          norm_num at hx
-          convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
-          norm_num
-        · intro x hx
-          norm_num at hx
-          rw [Pi.neg_apply, Left.nonneg_neg_iff]
-          exact (log_nonpos_iff hx.left).mpr hx.right.le
-      · have := tendsto_log_mul_rpow_nhds_zero zero_lt_one
-        simp_rw [rpow_one, mul_comm] at this
-        -- tendsto_nhdsWithin_of_tendsto_nhds should be under Tendsto namespace
-        convert this.sub (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id)
-        norm_num
-      · rw [(by simp : -1 = 1 * log 1 - 1)]
-        apply tendsto_nhdsWithin_of_tendsto_nhds
-        exact (continuousAt_id.mul (continuousAt_log one_ne_zero)).sub continuousAt_id
-    · -- ∫ x in 1..t, log x = t * log t + 1, just use integral_log_of_pos
-      rw [integral_log_of_pos zero_lt_one ht]
-      norm_num
-  · abel
-  · -- log is integrable on [[0, 1]]
-    rw [← neg_neg log]
-    apply IntervalIntegrable.neg
-    apply intervalIntegrable_deriv_of_nonneg (g := fun x ↦ -(x * log x - x))
-    · exact (continuous_mul_log.continuousOn.sub continuous_id.continuousOn).neg
-    · intro x hx
-      norm_num at hx
-      convert ((hasDerivAt_mul_log hx.left.ne.symm).sub (hasDerivAt_id x)).neg using 1
-      norm_num
-    · intro x hx
-      norm_num at hx
-      rw [Pi.neg_apply, Left.nonneg_neg_iff]
-      exact (log_nonpos_iff hx.left).mpr hx.right.le
-  · -- log is integrable on [[0, t]]
-    simp [Set.mem_uIcc, ht]
-
-
-lemma int₁₁ : ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) = 0 := by
-
-  sorry
-
-
-lemma int₁ :
-  ∫ x in (0)..(2 * π), log ‖circleMap 0 1 x - 1‖ = 0 := by
-
-  have {x : ℝ} : log ‖circleMap 0 1 x - 1‖ = log (4 * sin (x / 2) ^ 2) / 2 := by
-    dsimp [Complex.abs]
-    rw [log_sqrt (Complex.normSq_nonneg (circleMap 0 1 x - 1))]
-    congr
-    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
-    _ = 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]
-      ring
-    _ = 4 * sin (x / 2) ^ 2 := by
-      nth_rw 1 [← mul_one 4, ← sin_sq_add_cos_sq (x / 2)]
-      ring
-  simp_rw [this]
-  simp
-
-  have : ∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2) =  2 * ∫ (x : ℝ) in (0)..π, log (4 * sin x ^ 2) := by
-    have : 1 = 2 * (2 : ℝ)⁻¹ := by exact Eq.symm (mul_inv_cancel_of_invertible 2)
-    nth_rw 1 [← one_mul (∫ (x : ℝ) in (0)..2 * π, log (4 * sin (x / 2) ^ 2))]
-    rw [← mul_inv_cancel_of_invertible 2, mul_assoc]
-    let f := fun y ↦ log (4 * sin y ^ 2)
-    have {x : ℝ} : log (4 * sin (x / 2) ^ 2) = f (x / 2) := by simp
-    conv =>
-      left
-      right
-      right
-      arg 1
-      intro x
-      rw [this]
-    rw [intervalIntegral.inv_mul_integral_comp_div 2]
-    simp
-  rw [this]
-  simp
-  exact int₁₁