405 lines
12 KiB
Plaintext
405 lines
12 KiB
Plaintext
import Mathlib.Analysis.Analytic.Meromorphic
|
||
import Mathlib.Algebra.Order.AddGroupWithTop
|
||
import Nevanlinna.analyticAt
|
||
import Nevanlinna.mathlibAddOn
|
||
import Nevanlinna.meromorphicAt
|
||
|
||
|
||
open Topology
|
||
|
||
|
||
/- Strongly MeromorphicAt -/
|
||
def StronglyMeromorphicAt
|
||
(f : ℂ → ℂ)
|
||
(z₀ : ℂ) :=
|
||
(∀ᶠ (z : ℂ) in nhds z₀, f z = 0) ∨ (∃ (n : ℤ), ∃ g : ℂ → ℂ, (AnalyticAt ℂ g z₀) ∧ (g z₀ ≠ 0) ∧ (∀ᶠ (z : ℂ) in nhds z₀, f z = (z - z₀) ^ n • g z))
|
||
|
||
|
||
|
||
/- Strongly MeromorphicAt is Meromorphic -/
|
||
theorem StronglyMeromorphicAt.meromorphicAt
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : StronglyMeromorphicAt f z₀) :
|
||
MeromorphicAt f z₀ := by
|
||
rcases hf with h|h
|
||
· use 0; simp
|
||
rw [analyticAt_congr h]
|
||
exact analyticAt_const
|
||
· obtain ⟨n, g, h₁g, _, h₃g⟩ := h
|
||
rw [meromorphicAt_congr' h₃g]
|
||
apply MeromorphicAt.smul
|
||
apply MeromorphicAt.zpow
|
||
apply MeromorphicAt.sub
|
||
apply MeromorphicAt.id
|
||
apply MeromorphicAt.const
|
||
exact AnalyticAt.meromorphicAt h₁g
|
||
|
||
|
||
/- Strongly MeromorphicAt of non-negative order is analytic -/
|
||
theorem StronglyMeromorphicAt.analytic
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(h₁f : StronglyMeromorphicAt f z₀)
|
||
(h₂f : 0 ≤ h₁f.meromorphicAt.order):
|
||
AnalyticAt ℂ f z₀ := by
|
||
let h₁f' := h₁f
|
||
rcases h₁f' with h|h
|
||
· rw [analyticAt_congr h]
|
||
exact analyticAt_const
|
||
· obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h
|
||
rw [analyticAt_congr h₃g]
|
||
|
||
have : h₁f.meromorphicAt.order = n := by
|
||
rw [MeromorphicAt.order_eq_int_iff]
|
||
use g
|
||
constructor
|
||
· exact h₁g
|
||
· constructor
|
||
· exact h₂g
|
||
· exact Filter.EventuallyEq.filter_mono h₃g nhdsWithin_le_nhds
|
||
rw [this] at h₂f
|
||
apply AnalyticAt.smul
|
||
nth_rw 1 [← Int.toNat_of_nonneg (WithTop.coe_nonneg.mp h₂f)]
|
||
apply AnalyticAt.pow
|
||
apply AnalyticAt.sub
|
||
apply analyticAt_id -- Warning: want apply AnalyticAt.id
|
||
apply analyticAt_const -- Warning: want AnalyticAt.const
|
||
exact h₁g
|
||
|
||
|
||
/- Analytic functions are strongly meromorphic -/
|
||
theorem AnalyticAt.stronglyMeromorphicAt
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(h₁f : AnalyticAt ℂ f z₀) :
|
||
StronglyMeromorphicAt f z₀ := by
|
||
by_cases h₂f : h₁f.order = ⊤
|
||
· rw [AnalyticAt.order_eq_top_iff] at h₂f
|
||
tauto
|
||
· have : h₁f.order ≠ ⊤ := h₂f
|
||
rw [← ENat.coe_toNat_eq_self] at this
|
||
rw [eq_comm, AnalyticAt.order_eq_nat_iff] at this
|
||
right
|
||
use h₁f.order.toNat
|
||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := this
|
||
use g
|
||
tauto
|
||
|
||
|
||
/- Strong meromorphic depends only on germ -/
|
||
theorem stronglyMeromorphicAt_congr
|
||
{f g : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hfg : f =ᶠ[𝓝 z₀] g) :
|
||
StronglyMeromorphicAt f z₀ ↔ StronglyMeromorphicAt g z₀ := by
|
||
unfold StronglyMeromorphicAt
|
||
constructor
|
||
· intro h
|
||
rcases h with h|h
|
||
· left
|
||
exact Filter.EventuallyEq.rw h (fun x => Eq (g x)) (id (Filter.EventuallyEq.symm hfg))
|
||
· obtain ⟨n, h, h₁h, h₂h, h₃h⟩ := h
|
||
right
|
||
use n
|
||
use h
|
||
constructor
|
||
· assumption
|
||
· constructor
|
||
· assumption
|
||
· apply Filter.EventuallyEq.trans hfg.symm
|
||
assumption
|
||
· intro h
|
||
rcases h with h|h
|
||
· left
|
||
exact Filter.EventuallyEq.rw h (fun x => Eq (f x)) hfg
|
||
· obtain ⟨n, h, h₁h, h₂h, h₃h⟩ := h
|
||
right
|
||
use n
|
||
use h
|
||
constructor
|
||
· assumption
|
||
· constructor
|
||
· assumption
|
||
· apply Filter.EventuallyEq.trans hfg
|
||
assumption
|
||
|
||
|
||
theorem StronglyMeromorphicAt.order_eq_zero_iff
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : StronglyMeromorphicAt f z₀) :
|
||
hf.meromorphicAt.order = 0 ↔ f z₀ ≠ 0 := by
|
||
constructor
|
||
· intro h₁f
|
||
let A := hf.analytic (le_of_eq (id (Eq.symm h₁f)))
|
||
apply A.order_eq_zero_iff.1
|
||
let B := A.meromorphicAt_order
|
||
rw [h₁f] at B
|
||
apply WithTopCoe
|
||
rw [eq_comm]
|
||
exact B
|
||
· intro h
|
||
have hf' := hf
|
||
rcases hf with h₁|h₁
|
||
· have : f z₀ = 0 := by
|
||
apply Filter.EventuallyEq.eq_of_nhds h₁
|
||
tauto
|
||
· obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h₁
|
||
have : n = 0 := by
|
||
by_contra hContra
|
||
let A := Filter.EventuallyEq.eq_of_nhds h₃g
|
||
have : (0 : ℂ) ^ n = 0 := by
|
||
exact zero_zpow n hContra
|
||
simp at A
|
||
simp_rw [this] at A
|
||
simp at A
|
||
tauto
|
||
rw [this] at h₃g
|
||
simp at h₃g
|
||
|
||
have : hf'.meromorphicAt.order = 0 := by
|
||
apply (hf'.meromorphicAt.order_eq_int_iff 0).2
|
||
use g
|
||
constructor
|
||
· assumption
|
||
· constructor
|
||
· assumption
|
||
· simp
|
||
apply Filter.EventuallyEq.filter_mono h₃g
|
||
exact nhdsWithin_le_nhds
|
||
exact this
|
||
|
||
|
||
theorem StronglyMeromorphicAt.localIdentity
|
||
{f g : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : StronglyMeromorphicAt f z₀)
|
||
(hg : StronglyMeromorphicAt g z₀) :
|
||
f =ᶠ[𝓝[≠] z₀] g → f =ᶠ[𝓝 z₀] g := by
|
||
|
||
intro h
|
||
|
||
have t₀ : hf.meromorphicAt.order = hg.meromorphicAt.order := by
|
||
exact hf.meromorphicAt.order_congr h
|
||
|
||
by_cases cs : hf.meromorphicAt.order = 0
|
||
· rw [cs] at t₀
|
||
have h₁f := hf.analytic (le_of_eq (id (Eq.symm cs)))
|
||
have h₁g := hg.analytic (le_of_eq t₀)
|
||
exact h₁f.localIdentity h₁g h
|
||
· apply Mnhds h
|
||
let A := cs
|
||
rw [hf.order_eq_zero_iff] at A
|
||
simp at A
|
||
let B := cs
|
||
rw [t₀] at B
|
||
rw [hg.order_eq_zero_iff] at B
|
||
simp at B
|
||
rw [A, B]
|
||
|
||
|
||
|
||
|
||
|
||
/- Make strongly MeromorphicAt -/
|
||
noncomputable def MeromorphicAt.makeStronglyMeromorphicAt
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : MeromorphicAt f z₀) :
|
||
ℂ → ℂ := by
|
||
intro z
|
||
by_cases z = z₀
|
||
· by_cases h₁f : hf.order = (0 : ℤ)
|
||
· rw [hf.order_eq_int_iff] at h₁f
|
||
exact (Classical.choose h₁f) z₀
|
||
· exact 0
|
||
· exact f z
|
||
|
||
|
||
lemma m₁
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : MeromorphicAt f z₀) :
|
||
∀ z ≠ z₀, f z = hf.makeStronglyMeromorphicAt z := by
|
||
intro z hz
|
||
unfold MeromorphicAt.makeStronglyMeromorphicAt
|
||
simp [hz]
|
||
|
||
|
||
lemma m₂
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : MeromorphicAt f z₀) :
|
||
f =ᶠ[𝓝[≠] z₀] hf.makeStronglyMeromorphicAt := by
|
||
apply eventually_nhdsWithin_of_forall
|
||
exact fun x a => m₁ hf x a
|
||
|
||
|
||
theorem StronglyMeromorphicAt_of_makeStronglyMeromorphic
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : MeromorphicAt f z₀) :
|
||
StronglyMeromorphicAt hf.makeStronglyMeromorphicAt z₀ := by
|
||
|
||
by_cases h₂f : hf.order = ⊤
|
||
· have : hf.makeStronglyMeromorphicAt =ᶠ[𝓝 z₀] 0 := by
|
||
apply Mnhds
|
||
· apply Filter.EventuallyEq.trans (Filter.EventuallyEq.symm (m₂ hf))
|
||
exact (MeromorphicAt.order_eq_top_iff hf).1 h₂f
|
||
· unfold MeromorphicAt.makeStronglyMeromorphicAt
|
||
simp [h₂f]
|
||
|
||
apply AnalyticAt.stronglyMeromorphicAt
|
||
rw [analyticAt_congr this]
|
||
apply analyticAt_const
|
||
· let n := hf.order.untop h₂f
|
||
have : hf.order = n := by
|
||
exact Eq.symm (WithTop.coe_untop hf.order h₂f)
|
||
rw [hf.order_eq_int_iff] at this
|
||
obtain ⟨g, h₁g, h₂g, h₃g⟩ := this
|
||
right
|
||
use n
|
||
use g
|
||
constructor
|
||
· assumption
|
||
· constructor
|
||
· assumption
|
||
· apply Mnhds
|
||
· apply Filter.EventuallyEq.trans (Filter.EventuallyEq.symm (m₂ hf))
|
||
exact h₃g
|
||
· unfold MeromorphicAt.makeStronglyMeromorphicAt
|
||
simp
|
||
by_cases h₃f : hf.order = (0 : ℤ)
|
||
· let h₄f := (hf.order_eq_int_iff 0).1 h₃f
|
||
simp [h₃f]
|
||
obtain ⟨h₁G, h₂G, h₃G⟩ := Classical.choose_spec h₄f
|
||
simp at h₃G
|
||
have hn : n = 0 := Eq.symm ((fun {α} {a} {b} h => (WithTop.eq_untop_iff h).mpr) h₂f (id (Eq.symm h₃f)))
|
||
rw [hn]
|
||
rw [hn] at h₃g; simp at h₃g
|
||
simp
|
||
have : g =ᶠ[𝓝 z₀] (Classical.choose h₄f) := by
|
||
apply h₁g.localIdentity h₁G
|
||
exact Filter.EventuallyEq.trans (Filter.EventuallyEq.symm h₃g) h₃G
|
||
rw [Filter.EventuallyEq.eq_of_nhds this]
|
||
· have : hf.order ≠ 0 := h₃f
|
||
simp [this]
|
||
left
|
||
apply zero_zpow n
|
||
dsimp [n]
|
||
rwa [WithTop.untop_eq_iff h₂f]
|
||
|
||
|
||
theorem makeStronglyMeromorphic_id
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(hf : StronglyMeromorphicAt f z₀) :
|
||
f = hf.meromorphicAt.makeStronglyMeromorphicAt := by
|
||
|
||
funext z
|
||
by_cases hz : z = z₀
|
||
· rw [hz]
|
||
unfold MeromorphicAt.makeStronglyMeromorphicAt
|
||
simp
|
||
have h₀f := hf
|
||
rcases hf with h₁f|h₁f
|
||
· have A : f =ᶠ[𝓝[≠] z₀] 0 := by
|
||
apply Filter.EventuallyEq.filter_mono h₁f
|
||
exact nhdsWithin_le_nhds
|
||
let B := (MeromorphicAt.order_eq_top_iff h₀f.meromorphicAt).2 A
|
||
simp [B]
|
||
exact Filter.EventuallyEq.eq_of_nhds h₁f
|
||
· obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h₁f
|
||
rw [Filter.EventuallyEq.eq_of_nhds h₃g]
|
||
have : h₀f.meromorphicAt.order = n := by
|
||
rw [MeromorphicAt.order_eq_int_iff (StronglyMeromorphicAt.meromorphicAt h₀f) n]
|
||
use g
|
||
constructor
|
||
· assumption
|
||
· constructor
|
||
· assumption
|
||
· exact eventually_nhdsWithin_of_eventually_nhds h₃g
|
||
by_cases h₃f : h₀f.meromorphicAt.order = 0
|
||
· simp [h₃f]
|
||
have hn : n = (0 : ℤ) := by
|
||
rw [h₃f] at this
|
||
exact WithTop.coe_eq_zero.mp (id (Eq.symm this))
|
||
simp_rw [hn]
|
||
simp
|
||
let t₀ : h₀f.meromorphicAt.order = (0 : ℤ) := by
|
||
exact h₃f
|
||
let A := (h₀f.meromorphicAt.order_eq_int_iff 0).1 t₀
|
||
have : g =ᶠ[𝓝 z₀] (Classical.choose A) := by
|
||
obtain ⟨h₀, h₁, h₂⟩ := Classical.choose_spec A
|
||
apply h₁g.localIdentity h₀
|
||
rw [hn] at h₃g
|
||
simp at h₃g
|
||
simp at h₂
|
||
have h₄g : f =ᶠ[𝓝[≠] z₀] g := by
|
||
apply Filter.EventuallyEq.filter_mono h₃g
|
||
exact nhdsWithin_le_nhds
|
||
exact Filter.EventuallyEq.trans (Filter.EventuallyEq.symm h₄g) h₂
|
||
exact Filter.EventuallyEq.eq_of_nhds this
|
||
· simp [h₃f]
|
||
left
|
||
apply zero_zpow n
|
||
by_contra hn
|
||
rw [hn] at this
|
||
tauto
|
||
|
||
· exact m₁ (StronglyMeromorphicAt.meromorphicAt hf) z hz
|
||
|
||
|
||
theorem StronglyMeromorphicAt.decompose
|
||
{f : ℂ → ℂ}
|
||
{z₀ : ℂ}
|
||
(h₁f : StronglyMeromorphicAt f z₀)
|
||
(h₂f : h₁f.meromorphicAt.order ≠ ⊤) :
|
||
∃ g : ℂ → ℂ, (AnalyticAt ℂ g z₀)
|
||
∧ (g z₀ ≠ 0)
|
||
∧ (f = (fun z ↦ (z-z₀) ^ (h₁f.meromorphicAt.order.untop h₂f)) * g) := by
|
||
|
||
let n := - h₁f.meromorphicAt.order.untop h₂f
|
||
|
||
let g₁ := (fun z ↦ (z-z₀) ^ (-h₁f.meromorphicAt.order.untop h₂f)) * f
|
||
let g₁₁ := fun z ↦ (z-z₀) ^ (-h₁f.meromorphicAt.order.untop h₂f)
|
||
have h₁g₁₁ : MeromorphicAt g₁₁ z₀ := by
|
||
apply MeromorphicAt.zpow
|
||
apply AnalyticAt.meromorphicAt
|
||
apply AnalyticAt.sub
|
||
apply analyticAt_id
|
||
exact analyticAt_const
|
||
have h₂g₁₁ : h₁g₁₁.order = - h₁f.meromorphicAt.order.untop h₂f := by
|
||
rw [← WithTop.LinearOrderedAddCommGroup.coe_neg]
|
||
rw [h₁g₁₁.order_eq_int_iff]
|
||
use 1
|
||
constructor
|
||
· exact analyticAt_const
|
||
· constructor
|
||
· simp
|
||
· apply eventually_nhdsWithin_of_forall
|
||
simp [g₁₁]
|
||
have h₁g₁ : MeromorphicAt g₁ z₀ := h₁g₁₁.mul h₁f.meromorphicAt
|
||
have h₂g₁ : h₁g₁.order = 0 := by
|
||
rw [h₁g₁₁.order_mul h₁f.meromorphicAt]
|
||
rw [h₂g₁₁]
|
||
simp
|
||
rw [add_comm]
|
||
rw [LinearOrderedAddCommGroupWithTop.add_neg_cancel_of_ne_top h₂f]
|
||
let g := h₁g₁.makeStronglyMeromorphicAt
|
||
use g
|
||
have h₁g : StronglyMeromorphicAt g z₀ := by
|
||
exact StronglyMeromorphicAt_of_makeStronglyMeromorphic h₁g₁
|
||
have h₂g : h₁g.meromorphicAt.order = 0 := by
|
||
rw [← h₁g₁.order_congr (m₂ h₁g₁)]
|
||
exact h₂g₁
|
||
constructor
|
||
· apply analytic
|
||
· rw [h₂g]
|
||
· exact h₁g
|
||
· constructor
|
||
· exact (order_eq_zero_iff h₁g).mp h₂g
|
||
·
|
||
sorry
|