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Nevanlinna/firstMain.lean

@@ -88,10 +88,13 @@ theorem Nevanlinna_proximity₀
   exact Eq.symm (IsAbsoluteValue.abv_inv Norm.norm (f (circleMap 0 r x)))
   --
   apply MeromorphicOn.integrable_logpos_abs_f hr
+  intro z hx
+  exact h₁f z trivial
+  --
+  apply MeromorphicOn.integrable_logpos_abs_f hr
+  exact MeromorphicOn.inv_iff.mpr fun x a => h₁f x trivial
 
 
-  sorry
-
 theorem Nevanlinna_proximity
   {f : ℂ → ℂ}
   {r : ℝ}
@@ -105,9 +108,8 @@ theorem Nevanlinna_proximity
     rw [← mul_sub]; congr
     exact loglogpos
 
-
+  by_cases h₂r : 0 < r
-
+  · exact Nevanlinna_proximity₀ h₁f h₂r
-  have hr : 0 < r := by sorry
 
   unfold MeromorphicOn.m_infty
   rw [← mul_sub]; congr
@@ -116,8 +118,10 @@ theorem Nevanlinna_proximity
   simp_rw [loglogpos]; congr
   exact Eq.symm (IsAbsoluteValue.abv_inv Norm.norm (f (circleMap 0 r x)))
   --
-  apply MeromorphicOn.integrable_logpos_abs_f hr
+  apply MeromorphicOn.integrable_logpos_abs_f
 
+
+  sorry
   sorry
 
 noncomputable def MeromorphicOn.T_infty

Nevanlinna/intervalIntegrability.lean

@@ -1,13 +1,5 @@
-import Mathlib.Analysis.Analytic.Meromorphic
 import Mathlib.MeasureTheory.Integral.CircleIntegral
-import Mathlib.MeasureTheory.Integral.IntervalIntegral
+import Nevanlinna.periodic_integrability
-import Nevanlinna.analyticAt
-import Nevanlinna.divisor
-import Nevanlinna.meromorphicAt
-import Nevanlinna.meromorphicOn_divisor
-import Nevanlinna.stronglyMeromorphicOn
-import Nevanlinna.stronglyMeromorphicOn_eliminate
-import Nevanlinna.mathlibAddOn
 
 open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
@@ -96,3 +88,61 @@ theorem integral_congr_changeDiscrete
   constructor
   · apply Set.Countable.measure_zero (d hr hU hf)
   · tauto
+
+
+lemma circleMap_neg
+  {r x : ℝ} :
+  circleMap 0 (-r) x = circleMap 0 r (x + π) := by
+  unfold circleMap
+  simp
+  rw [add_mul]
+  rw [Complex.exp_add]
+  simp
+
+
+theorem integrability_congr_negRadius
+  {f : ℂ → ℝ}
+  {r : ℝ} :
+  IntervalIntegrable (fun (θ : ℝ) ↦ f (circleMap 0   r  θ)) MeasureTheory.volume 0 (2 * π) →
+  IntervalIntegrable (fun (θ : ℝ) ↦ f (circleMap 0 (-r) θ)) MeasureTheory.volume 0 (2 * π) := by
+
+  intro h
+  simp_rw [circleMap_neg]
+
+  let A := IntervalIntegrable.comp_add_right h π
+
+  have t₀ : Function.Periodic (fun (θ : ℝ) ↦ f (circleMap 0   r  θ)) (2 * π) := by
+    intro x
+    simp
+    congr 1
+    exact periodic_circleMap 0 r x
+  rw [← zero_add (2 * π)] at h
+  have B := periodic_integrability4 (a₁ := π) (a₂ := 3 * π) t₀ two_pi_pos h
+  let A := IntervalIntegrable.comp_add_right B π
+  simp at A
+  have : 3 * π - π = 2 * π := by
+    ring
+  rw [this] at A
+  exact A
+
+
+theorem integrabl_congr_negRadius
+  {f : ℂ → ℝ}
+  {r : ℝ} :
+  ∫ (x : ℝ) in (0)..(2 * π), f (circleMap 0 r x) = ∫ (x : ℝ) in (0)..(2 * π), f (circleMap 0 (-r) x) := by
+
+  simp_rw [circleMap_neg]
+  have t₀ : Function.Periodic (fun (θ : ℝ) ↦ f (circleMap 0   r  θ)) (2 * π) := by
+    intro x
+    simp
+    congr 1
+    exact periodic_circleMap 0 r x
+  --rw [← zero_add (2 * π)] at h
+  --have B := periodic_integrability4 (a₁ := π) (a₂ := 3 * π) t₀ two_pi_pos h
+  have B := intervalIntegral.integral_comp_add_right (a := 0) (b := 2 * π) (fun θ => f (circleMap 0 r θ)) π
+  rw [B]
+  let X := t₀.intervalIntegral_add_eq 0 (0 + π)
+  simp at X
+  rw [X]
+  simp
+  rw [add_comm]

Nevanlinna/meromorphicOn_integrability.lean

@@ -16,9 +16,10 @@ open scoped Interval Topology
 open Real Filter MeasureTheory intervalIntegral
 
 
-theorem MeromorphicOn.integrable_log_abs_f
+theorem MeromorphicOn.integrable_log_abs_f₀
   {f : ℂ → ℂ}
   {r : ℝ}
+  -- WARNING: Not optimal. This needs to go
   (hr : 0 < r)
   -- WARNING: Not optimal. It suffices to be meromorphic on the Sphere
   (h₁f : MeromorphicOn f (Metric.closedBall (0 : ℂ) r)) :
@@ -111,10 +112,51 @@ theorem MeromorphicOn.integrable_log_abs_f
     rw [integrability_congr_changeDiscrete hU this]
 
     have : ∀ x ∈ Metric.closedBall 0 r, F x = 0 := by
-      sorry
+      intro x hx
+      let A := h₂f ⟨x, hx⟩
+      rw [← makeStronglyMeromorphicOn_changeOrder h₁f hx] at A
+      let B := ((stronglyMeromorphicOn_of_makeStronglyMeromorphicOn h₁f) x hx).order_eq_zero_iff.not.1
+      simp [A] at B
+      assumption
+    have : (fun z => log ‖F z‖) ∘ circleMap 0 r = 0 := by
+      funext x
+      simp
+      left
+      apply this
+      simp [abs_of_pos hr]
+    simp_rw [this]
+    apply intervalIntegral.intervalIntegrable_const
 
 
+theorem MeromorphicOn.integrable_log_abs_f
+  {f : ℂ → ℂ}
+  {r : ℝ}
+  -- WARNING: Not optimal. It suffices to be meromorphic on the Sphere
+  (h₁f : MeromorphicOn f (Metric.closedBall (0 : ℂ) |r|)) :
+  IntervalIntegrable (fun z ↦ log ‖f (circleMap 0 r z)‖) MeasureTheory.volume 0 (2 * π) := by
+
+  by_cases h₁r : r = 0
+  · rw [h₁r]
+    simp
+  · by_cases h₂r : 0 < r
+    · have : |r| = r := by
+        exact abs_of_pos h₂r
+      rw [this] at h₁f
+      exact MeromorphicOn.integrable_log_abs_f₀ h₂r h₁f
+    · have t₀ : 0 < -r := by
         sorry
+      have : |r| = -r := by
+        apply abs_of_neg
+        exact Left.neg_pos_iff.mp t₀
+      rw [this] at h₁f
+      let A := MeromorphicOn.integrable_log_abs_f₀ t₀ h₁f
+
+      let B := integrability_congr_negRadius (f := fun z => log ‖f z‖) (r := -r)
+      let C := B A
+      simp at C
+      simpa
+
+
 
 theorem MeromorphicOn.integrable_logpos_abs_f
   {f : ℂ → ℂ}