62 lines
1.4 KiB
Plaintext
62 lines
1.4 KiB
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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open scoped Interval Topology
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open Real Filter MeasureTheory intervalIntegral
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noncomputable def MeromorphicOn.divisor
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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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Divisor U where
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toFun := by
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intro z
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if hz : z ∈ U then
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exact ((hf z hz).order.untop' 0 : ℤ)
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else
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exact 0
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supportInU := by
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intro z hz
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simp at hz
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by_contra h₂z
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simp [h₂z] at hz
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locallyFiniteInU := by
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intro z hz
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apply eventually_nhdsWithin_iff.2
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rw [eventually_nhds_iff]
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rcases AnalyticAt.eventually_eq_zero_or_eventually_ne_zero (hf z hz) with h|h
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· rw [eventually_nhds_iff] at h
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
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use N
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constructor
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· intro y h₁y _
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by_cases h₃y : y ∈ U
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· simp [h₃y]
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right
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rw [AnalyticAt.order_eq_top_iff (hf y h₃y)]
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rw [eventually_nhds_iff]
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use N
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· simp [h₃y]
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· tauto
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· rw [eventually_nhdsWithin_iff, eventually_nhds_iff] at h
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obtain ⟨N, h₁N, h₂N, h₃N⟩ := h
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use N
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constructor
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· intro y h₁y h₂y
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by_cases h₃y : y ∈ U
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· simp [h₃y]
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left
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rw [AnalyticAt.order_eq_zero_iff (hf y h₃y)]
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exact h₁N y h₁y h₂y
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· simp [h₃y]
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· tauto
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