The Casorati-Weierstrass Theorem states that if a function is holomorphic on a punctured neighborhood of an essential singularity, then in every such neighborhood the function's values come arbitrarily close to every complex number, so its image is dense in the complex plane. Named for Felice Casorati and Karl Weierstrass, it describes just how wildly a complex function behaves near an essential singularity, a picture later sharpened considerably by Picard's Theorem.
Facts
StatementIf f is holomorphic with an essential singularity at z0, then for any neighborhood V of z0, f of V minus z0 is dense in the complex plane. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Casorati-Weierstrass theorem (Wikipedia)
Proved By
Sources
1. Casorati-Weierstrass theorem (Wikipedia)
Formal Statement
if V is any neighborhood of zā contained in U, then f(V ā {zā}) is dense in ā.
- In Branch: Complex Analysis, Lead sentence
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