24 lines
687 B
Plaintext
24 lines
687 B
Plaintext
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import Mathlib.Analysis.Analytic.Meromorphic
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import Nevanlinna.analyticAt
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import Nevanlinna.divisor
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import Nevanlinna.meromorphicAt
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import Nevanlinna.meromorphicOn_divisor
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import Nevanlinna.stronglyMeromorphicOn
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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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theorem MeromorphicOn.decompose
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{f : ℂ → ℂ}
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{U : Set ℂ}
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(h₁U : IsConnected U)
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(h₂U : IsCompact U)
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(h₁f : MeromorphicOn f U)
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(h₂f : ∃ z₀ ∈ U, f z₀ ≠ 0) :
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∃ g : ℂ → ℂ, (AnalyticOnNhd ℂ g U)
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∧ (∀ z ∈ U, g z ≠ 0)
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∧ (Set.EqOn h₁f.makeStronglyMeromorphicOn (fun z ↦ ∏ᶠ p ∈ h₁f.divisor.support, (z-p) ) U) := by
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sorry
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