Update holomorphic_zero.lean

Nevanlinna/holomorphic_zero.lean

@@ -1,10 +1,36 @@
+import Mathlib.Analysis.Analytic.IsolatedZeros
 import Nevanlinna.holomorphic
 
 
-def zeroDivisor
-  {f : ℂ → ℂ}
-  {R : ℝ}
-  (h₁f : ∀ z ∈ Metric.closedBall z R, HolomorphicAt f z)
-  (h₂f : ∃ z ∈ Metric.closedBall z R, f z ≠ 0) :
+noncomputable def zeroDivisor
+  (f : ℂ → ℂ) :
   ℂ → ℕ := by
+  intro z
+  if hf : AnalyticAt ℂ f z then
+    exact hf.order.toNat
+  else
+    exact 0
+
+
+theorem discreteZeros
+  {f : ℂ → ℂ} :
+  DiscreteTopology (Function.support (zeroDivisor f)) := by
+  sorry
+
+
+theorem zeroDivisor_finiteOnCompact
+  {f : ℂ → ℂ}
+  {s : Set ℂ}
+  (hs : IsCompact s) :
+  Set.Finite (s ∩ Function.support (zeroDivisor f)) := by
+  sorry
+
+
+theorem eliminatingZeros
+  {f : ℂ → ℂ}
+  {z₀ : ℂ}
+  {R : ℝ}
+  (h₁f : ∀ z ∈ Metric.ball z₀ R, HolomorphicAt f z)
+  (h₂f : ∃ z ∈ Metric.ball z₀ R, f z ≠ 0) :
+  ∃ F : ℂ → ℂ, ∀ z ∈ Metric.ball z₀ R, (HolomorphicAt F z) ∧ (f z = (F z) * ∏ᶠ a ∈ Metric.ball z₀ R, (z - a) ^ (zeroDivisor f a) ) := by
   sorry