Theorems · Theorem · complex analysis

meromorphicOrderAt_eq_int_iff

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {f : 𝕜 → E} {x : 𝕜} {n : ℤ},
  MeromorphicAt f x →
    (meromorphicOrderAt f x = ↑n ↔
      ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in nhdsWithin x {x}ᶜ, f z = (z - x) ^ n • g z)

The order of a meromorphic function f at z₀ equals an integer n iff f can locally be written as f z = (z - z₀) ^ n • g z, where g is analytic and does not vanish at z₀.

Defined in

Mathlib.Analysis.Meromorphic.Order

Cited by

31 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace