75 lines
2.4 KiB
Plaintext
75 lines
2.4 KiB
Plaintext
import Mathlib.Analysis.Analytic.Meromorphic
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import Mathlib.Analysis.Calculus.ContDiff.Basic
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import Mathlib.Analysis.Calculus.FDeriv.Add
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import Nevanlinna.analyticAt
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variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
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variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
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variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
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variable {G : Type*} [NormedAddCommGroup G] [NormedSpace 𝕜 G]
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variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace 𝕜 G']
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variable {f f₀ f₁ g : E → F}
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variable {f' f₀' f₁' g' : E →L[𝕜] F}
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variable (e : E →L[𝕜] F)
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variable {x : E}
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variable {s t : Set E}
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variable {L L₁ L₂ : Filter E}
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variable {R : Type*} [Semiring R] [Module R F] [SMulCommClass 𝕜 R F] [ContinuousConstSMul R F]
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-- import Mathlib.Analysis.Calculus.FDeriv.Add
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@[fun_prop]
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theorem Differentiable.const_smul' (h : Differentiable 𝕜 f) (c : R) :
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Differentiable 𝕜 (c • f) := by
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have : c • f = fun x ↦ c • f x := rfl
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rw [this]
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exact Differentiable.const_smul h c
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-- Mathlib.Analysis.Calculus.ContDiff.Basic
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theorem ContDiff.const_smul' {f : E → F} (c : R) (hf : ContDiff 𝕜 n f) :
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ContDiff 𝕜 n (c • f) := by
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have : c • f = fun x ↦ c • f x := rfl
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rw [this]
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exact ContDiff.const_smul c hf
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-- Mathlib.Analysis.Analytic.Meromorphic
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theorem meromorphicAt_congr
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{𝕜 : Type u_1} [NontriviallyNormedField 𝕜]
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{E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
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{f : 𝕜 → E} {g : 𝕜 → E} {x : 𝕜}
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(h : f =ᶠ[nhdsWithin x {x}ᶜ] g) : MeromorphicAt f x ↔ MeromorphicAt g x :=
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⟨fun hf ↦ hf.congr h, fun hg ↦ hg.congr h.symm⟩
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theorem meromorphicAt_congr'
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{𝕜 : Type u_1} [NontriviallyNormedField 𝕜]
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{E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
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{f : 𝕜 → E} {g : 𝕜 → E} {x : 𝕜}
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(h : f =ᶠ[nhds x] g) : MeromorphicAt f x ↔ MeromorphicAt g x :=
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meromorphicAt_congr (Filter.EventuallyEq.filter_mono h nhdsWithin_le_nhds)
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open Topology Filter
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lemma Mnhds
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{α : Type}
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{f g : ℂ → α}
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{z₀ : ℂ}
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(h₁ : f =ᶠ[𝓝[≠] z₀] g)
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(h₂ : f z₀ = g z₀) :
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f =ᶠ[𝓝 z₀] g := by
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apply eventually_nhds_iff.2
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obtain ⟨t, h₁t, h₂t⟩ := eventually_nhds_iff.1 (eventually_nhdsWithin_iff.1 h₁)
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use t
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constructor
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· intro y hy
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by_cases h₂y : y ∈ ({z₀}ᶜ : Set ℂ)
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· exact h₁t y hy h₂y
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· simp at h₂y
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rwa [h₂y]
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· exact h₂t
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