118 lines
3.5 KiB
Plaintext
118 lines
3.5 KiB
Plaintext
import Init.Classical
|
||
import Mathlib.Analysis.Analytic.Meromorphic
|
||
import Mathlib.Topology.ContinuousOn
|
||
import Mathlib.Analysis.Analytic.IsolatedZeros
|
||
import Nevanlinna.holomorphic
|
||
|
||
|
||
|
||
theorem AnalyticOn.order_eq_nat_iff
|
||
{f : ℂ → ℂ}
|
||
{U : Set ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : AnalyticOn ℂ f U)
|
||
(hz₀ : z₀ ∈ U)
|
||
(n : ℕ) :
|
||
(hf z₀ hz₀).order = ↑n ↔ ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ g z₀ ≠ 0 ∧ ∀ z, f z = (z - z₀) ^ n • g z := by
|
||
|
||
constructor
|
||
-- Direction →
|
||
intro hn
|
||
obtain ⟨gloc, h₁gloc, h₂gloc, h₃gloc⟩ := (AnalyticAt.order_eq_nat_iff (hf z₀ hz₀) n).1 hn
|
||
|
||
-- Define a candidate function; this is (f z) / (z - z₀) ^ n with the
|
||
-- removable singularity removed
|
||
let g : ℂ → ℂ := fun z ↦ if z = z₀ then gloc z₀ else (f z) / (z - z₀) ^ n
|
||
|
||
-- Describe g near z₀
|
||
have g_near_z₀ : ∀ᶠ (z : ℂ) in nhds z₀, g z = gloc z := by
|
||
rw [eventually_nhds_iff]
|
||
obtain ⟨t, h₁t, h₂t, h₃t⟩ := eventually_nhds_iff.1 h₃gloc
|
||
use t
|
||
constructor
|
||
· intro y h₁y
|
||
by_cases h₂y : y = z₀
|
||
· dsimp [g]; simp [h₂y]
|
||
· dsimp [g]; simp [h₂y]
|
||
rw [div_eq_iff_mul_eq, eq_comm, mul_comm]
|
||
exact h₁t y h₁y
|
||
norm_num
|
||
rw [sub_eq_zero]
|
||
tauto
|
||
· constructor
|
||
· assumption
|
||
· assumption
|
||
|
||
-- Describe g near points z₁ that are different from z₀
|
||
have g_near_z₁ {z₁ : ℂ} : z₁ ≠ z₀ → ∀ᶠ (z : ℂ) in nhds z₁, g z = f z / (z - z₀) ^ n := by
|
||
intro hz₁
|
||
rw [eventually_nhds_iff]
|
||
use {z₀}ᶜ
|
||
constructor
|
||
· intro y hy
|
||
simp at hy
|
||
simp [g, hy]
|
||
· exact ⟨isOpen_compl_singleton, hz₁⟩
|
||
|
||
-- Use g and show that it has all required properties
|
||
use g
|
||
constructor
|
||
· -- AnalyticOn ℂ g U
|
||
intro z h₁z
|
||
by_cases h₂z : z = z₀
|
||
· rw [h₂z]
|
||
apply AnalyticAt.congr h₁gloc
|
||
exact Filter.EventuallyEq.symm g_near_z₀
|
||
· simp_rw [eq_comm] at g_near_z₁
|
||
apply AnalyticAt.congr _ (g_near_z₁ h₂z)
|
||
apply AnalyticAt.div
|
||
exact hf z h₁z
|
||
apply AnalyticAt.pow
|
||
apply AnalyticAt.sub
|
||
apply analyticAt_id
|
||
apply analyticAt_const
|
||
simp
|
||
rw [sub_eq_zero]
|
||
tauto
|
||
· constructor
|
||
· simp [g]; tauto
|
||
· intro z
|
||
by_cases h₂z : z = z₀
|
||
· rw [h₂z, g_near_z₀.self_of_nhds]
|
||
exact h₃gloc.self_of_nhds
|
||
· rw [(g_near_z₁ h₂z).self_of_nhds]
|
||
simp [h₂z]
|
||
rw [div_eq_mul_inv, mul_comm, mul_assoc, inv_mul_cancel]
|
||
simp; norm_num
|
||
rw [sub_eq_zero]
|
||
tauto
|
||
|
||
-- direction ←
|
||
intro h
|
||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := h
|
||
rw [AnalyticAt.order_eq_nat_iff]
|
||
use g
|
||
exact ⟨h₁g z₀ hz₀, ⟨h₂g, Filter.eventually_of_forall h₃g⟩⟩
|
||
|
||
|
||
theorem AnalyticOn.order_eq_nat_iff'
|
||
{f : ℂ → ℂ}
|
||
{U : Set ℂ}
|
||
{A : Finset U}
|
||
(hf : AnalyticOn ℂ f U)
|
||
(n : ℂ → ℕ) :
|
||
(∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a : A, g a ≠ 0) ∧ ∀ z, f z = (∏ a : A, (z - a) ^ (n a)) • g z := by
|
||
|
||
apply Finset.induction (α := U) (p := fun A ↦ (∀ a ∈ A, (hf a.1 a.2).order = n a) → ∃ (g : ℂ → ℂ), AnalyticOn ℂ g U ∧ (∀ a : A, g a ≠ 0) ∧ ∀ z, f z = (∏ a : A, (z - a) ^ (n a)) • g z)
|
||
|
||
-- case empty
|
||
simp
|
||
use f
|
||
simp
|
||
exact hf
|
||
|
||
-- case insert
|
||
intro b₀ B hb iHyp
|
||
intro hBinsert
|
||
obtain ⟨g₀, h₁g₀, h₂g₀, h₃g₀⟩ := iHyp (fun a ha ↦ hBinsert a (Finset.mem_insert_of_mem ha))
|