104 lines
2.9 KiB
Plaintext
104 lines
2.9 KiB
Plaintext
import Mathlib.Analysis.Analytic.Meromorphic
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import Nevanlinna.analyticAt
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import Nevanlinna.mathlibAddOn
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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, _, h₃g⟩ := h
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rw [meromorphicAt_congr' h₃g]
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apply MeromorphicAt.smul
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apply MeromorphicAt.zpow
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apply MeromorphicAt.sub
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apply MeromorphicAt.id
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apply MeromorphicAt.const
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exact AnalyticAt.meromorphicAt h₁g
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/- Strongly MeromorphicAt of non-negative 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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let h₁f' := h₁f
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rcases h₁f' with h|h
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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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rw [analyticAt_congr h₃g]
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have : h₁f.meromorphicAt.order = n := by
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rw [MeromorphicAt.order_eq_int_iff]
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use g
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constructor
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· exact h₁g
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· constructor
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· exact h₂g
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· exact Filter.EventuallyEq.filter_mono h₃g nhdsWithin_le_nhds
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rw [this] at h₂f
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apply AnalyticAt.smul
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nth_rw 1 [← Int.toNat_of_nonneg (WithTop.coe_nonneg.mp h₂f)]
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apply AnalyticAt.pow
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apply AnalyticAt.sub
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apply analyticAt_id -- Warning: want apply AnalyticAt.id
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apply analyticAt_const -- Warning: want AnalyticAt.const
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exact h₁g
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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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by_cases h₂f : hf.order = ⊤
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· exact 0
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· have : ∃ n : ℤ, hf.order = n := by
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exact Option.ne_none_iff_exists'.mp h₂f
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let o := hf.order.untop h₂f
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have : hf.order = o := by
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exact Eq.symm (WithTop.coe_untop hf.order h₂f)
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rw [MeromorphicAt.order_eq_int_iff] at this
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obtain ⟨g, hg⟩ := this
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exact fun z ↦ (z - z₀) ^ o • g z
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sorry
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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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