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 : ℂ → ℂ}
|
|
|
|
|
{U : Set ℂ}
|
|
|
|
|
(h₁U : IsConnected U)
|
|
|
|
|
(h₂U : IsCompact U)
|
|
|
|
|
(h₁f : MeromorphicOn f U)
|
|
|
|
|
(h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
|
|
|
|
|
∃ g : ℂ → ℂ, (AnalyticOnNhd ℂ g U)
|
|
|
|
|
∧ (∀ z ∈ U, g z ≠ 0)
|
2024-11-07 16:10:37 +01:00
|
|
|
|
∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn (fun z ↦ ∏ᶠ p, (z - p) ^ (h₁f.divisor p) * g z ) U) := by
|
|
|
|
|
|
|
|
|
|
let g₁ : ℂ → ℂ := 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 ℂ 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
|