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Nevanlinna/harmonicAt_meanValue.lean

@@ -7,8 +7,7 @@ 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

@@ -1,54 +0,0 @@
-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

@@ -0,0 +1,162 @@
+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