Theorems · Theorem · complex analysis

MeromorphicOn.exists_meromorphicOrderAt_ne_top_iff_forall

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {f : 𝕜 → E} {U : Set 𝕜},
  MeromorphicOn f U → IsConnected U → ((∃ u, meromorphicOrderAt f ↑u ≠ ⊤) ↔ ∀ (u : ↑U), meromorphicOrderAt 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. See Meromorphic.exists_meromorphicOrderAt_ne_top_iff_forall in file Mathlib/Analysis/Meromorphic/RCLike for a related result assuming that f is meromorphic on all of 𝕜.

Defined in

Mathlib.Analysis.Meromorphic.Order

Cited by

7 results in Mathlib

Foundations

Depth 199 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpace