71 lines
1.8 KiB
Plaintext
71 lines
1.8 KiB
Plaintext
import Mathlib.Analysis.Analytic.Meromorphic
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import Nevanlinna.analyticAt
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/- Strongly MeromorphicAt -/
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def StronglyMeromorphicAt
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(f : ℂ → ℂ)
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(z₀ : ℂ) :=
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(∀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))
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/- Strongly MeromorphicAt is Meromorphic -/
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theorem StronglyMeromorphicAt.meromorphicAt
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(hf : StronglyMeromorphicAt f z₀) :
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MeromorphicAt f z₀ := by
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rcases hf with h|h
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· use 0; simp
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rw [analyticAt_congr h]
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exact analyticAt_const
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· obtain ⟨n, g, h₁g, h₂g, h₃g⟩ := h
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have : MeromorphicAt (fun z ↦ (z - z₀) ^ n • g z) z₀ := by
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simp
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apply MeromorphicAt.mul
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apply MeromorphicAt.zpow
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apply MeromorphicAt.sub
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sorry
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apply MeromorphicAt.congr this
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rw [Filter.eventuallyEq_comm]
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exact Filter.EventuallyEq.filter_mono h₃g nhdsWithin_le_nhds
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/- Strongly MeromorphicAt of positive order is analytic -/
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theorem StronglyMeromorphicAt.analytic
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(h₁f : StronglyMeromorphicAt f z₀)
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(h₂f : 0 ≤ h₁f.meromorphicAt.order):
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AnalyticAt ℂ f z₀ := by
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sorry
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/- Make strongly MeromorphicAt -/
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def MeromorphicAt.makeStronglyMeromorphicAt
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(hf : MeromorphicAt f z₀) :
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ℂ → ℂ := by
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exact 0
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theorem StronglyMeromorphicAt_of_makeStronglyMeromorphic
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(hf : MeromorphicAt f z₀) :
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StronglyMeromorphicAt hf.makeStronglyMeromorphic z₀ := by
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sorry
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theorem makeStronglyMeromorphic_eventuallyEq
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{f : ℂ → ℂ}
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{z₀ : ℂ}
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(hf : MeromorphicAt f z₀) :
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∀ᶠ (z : ℂ) in nhdsWithin z₀ {z₀}ᶜ, f z = hf.makeStronglyMeromorphicAt z := by
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sorry
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