Theorems · Theorem · complex analysis

MeromorphicAt.eventually_analyticAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {x : 𝕜} [CompleteSpace E] {f : 𝕜 → E},
  MeromorphicAt f x → ∀ᶠ (y : 𝕜) in nhdsWithin x {x}ᶜ, AnalyticAt 𝕜 f y

In a complete space, a function which is meromorphic at a point is analytic at all nearby points. The completeness assumption can be dispensed with if one assumes that f is meromorphic on a set around x, see MeromorphicOn.eventually_analyticAt.

Defined in

Mathlib.Analysis.Meromorphic.Basic

Cited by

1 results in Mathlib

Foundations

Depth 192 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceCompleteSpace