nevanlinna/Nevanlinna/meromorphicOn_decompose.lean

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2024-11-07 12:08:52 +01:00
import Mathlib.Analysis.Analytic.Meromorphic
import Nevanlinna.analyticAt
import Nevanlinna.divisor
import Nevanlinna.meromorphicAt
import Nevanlinna.meromorphicOn_divisor
import Nevanlinna.stronglyMeromorphicOn
open scoped Interval Topology
open Real Filter MeasureTheory intervalIntegral
theorem MeromorphicOn.decompose
{f : }
{U : Set }
(h₁U : IsConnected U)
(h₂U : IsCompact U)
(h₁f : MeromorphicOn f U)
(h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
∃ g : , (AnalyticOnNhd g U)
∧ (∀ z ∈ U, g z ≠ 0)
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∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p) * g z ) U) := by
let g₁ : := f * (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p))
have h₁g₁ : MeromorphicOn g₁ U := by sorry
let g := h₁g₁.makeStronglyMeromorphicOn
have h₁g : MeromorphicOn g U := by sorry
have h₂g : ∀ z : U, (h₁g z.1 z.2).order = 0 := by sorry
have h₃g : StronglyMeromorphicOn g U := by sorry
have h₄g : AnalyticOnNhd g U := by
intro z hz
apply StronglyMeromorphicAt.analytic (h₃g z hz)
rw [h₂g ⟨z, hz⟩]
use g
constructor
· exact h₄g
· constructor
· intro z hz
rw [← (h₄g z hz).order_eq_zero_iff]
let A := (h₄g z hz).meromorphicAt_order
let B := h₂g ⟨z, hz⟩
sorry
· intro z hz
sorry