Theorems · Theorem · complex analysis

MeromorphicOn.exists_ecanonicalDecomp

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℂ E] {R : ℝ} {f : ℂ → E},
  MeromorphicOn f (Metric.closedBall 0 R) →
    (∀ (u : ↑(Metric.closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤) → ∃ h, Complex.ECanonicalDecomp f h R

Extended canonical decomposition: A meromorphic function on a closed disk is equal, up to modification over a discrete set, to a product of a non-vanishing analytic function, canonical factors and meromorphic functions of the form (x - const) ^ n where const is on the circumference of the disk.

Defined in

Mathlib.Analysis.Complex.CanonicalDecomposition

Cited by

0 results in Mathlib

Foundations

Depth 213 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpace