66 lines
1.5 KiB
Plaintext
66 lines
1.5 KiB
Plaintext
import Nevanlinna.stronglyMeromorphicAt
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open Topology
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/- Strongly MeromorphicOn -/
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def StronglyMeromorphicOn
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(f : ℂ → ℂ)
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(U : Set ℂ) :=
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∀ z ∈ U, StronglyMeromorphicAt f z
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/- Strongly MeromorphicAt is Meromorphic -/
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theorem StronglyMeromorphicOn.meromorphicOn
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hf : StronglyMeromorphicOn f U) :
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MeromorphicOn f U := by
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intro z hz
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exact StronglyMeromorphicAt.meromorphicAt (hf z hz)
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/- Strongly MeromorphicOn of non-negative order is analytic -/
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theorem StronglyMeromorphicOn.analytic
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁f : StronglyMeromorphicOn f U)
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(h₂f : ∀ x, (hx : x ∈ U) → 0 ≤ (h₁f x hx).meromorphicAt.order):
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∀ z ∈ U, AnalyticAt ℂ f z := by
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intro z hz
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apply StronglyMeromorphicAt.analytic
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exact h₂f z hz
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exact h₁f z hz
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/- Analytic functions are strongly meromorphic -/
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theorem AnalyticOn.stronglyMeromorphicOn
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁f : AnalyticOnNhd ℂ f U) :
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StronglyMeromorphicOn f U := by
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intro z hz
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apply AnalyticAt.stronglyMeromorphicAt
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exact h₁f z hz
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/- Make strongly MeromorphicAt -/
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noncomputable def MeromorphicOn.makeStronglyMeromorphicOn
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hf : MeromorphicOn f U) :
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ℂ → ℂ := by
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intro z
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by_cases hz : z ∈ U
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· exact (hf z hz).makeStronglyMeromorphicAt z
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· exact f z
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theorem StronglyMeromorphicOn_of_makeStronglyMeromorphicOn
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(hf : MeromorphicOn f U) :
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StronglyMeromorphicOn hf.makeStronglyMeromorphicOn U := by
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sorry
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