Theorems · Theorem · complex analysis

AnalyticOnNhd.exists_analyticOrderAt_ne_top_iff_forall

∀ {𝕜 : Type u_1} {E : Type u_2} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {U : Set 𝕜} {f : 𝕜 → E},
  AnalyticOnNhd 𝕜 f U → IsConnected U → ((∃ u, analyticOrderAt f ↑u ≠ ⊤) ↔ ∀ (u : ↑U), analyticOrderAt f ↑u ≠ ⊤)

On a connected set, there exists a point where a meromorphic function f has finite order iff f has finite order at every point.

Defined in

Mathlib.Analysis.Analytic.Order

Cited by

1 results in Mathlib

Foundations

Depth 200 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace