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 Proof Method Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Casorati-Weierstrass theorem (Wikipedia)
Principal Result Of
Source Felice Casorati (MacTutor)
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.
Lead section
In Russian literature it is called Sokhotski's theorem, because it was discovered independently by Sokhotski in 1868.
Principal Result Of: Karl Weierstrass, Lead sentence of the article on the result
It is named for Karl Theodor Wilhelm Weierstrass and Felice Casorati.
- In Branch: Complex Analysis, Lead sentence
View the Source2. ProofWiki: Weierstrass-Casorati Theorem
ProofWikiProof sectionQuote, Proof section
Aiming for a contradiction, suppose $\exists s < 1$ such that:
Felice Casorati (MacTutor)
Principal Result Of: Felice Casorati (mathematician), Biography, best rememberedQuote, Principal Result Of: Felice Casorati (mathematician), Biography, best remembered
Casorati is best remembered for the Casorati- Weierstrass theorem
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.