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Casorati-Weierstrass theorem (Theorem)

Given a domain $ U\subset\mathbb{C}$, $ a\in U$, and $ f:U\setminus\{a\}\to\mathbb{C}$ being holomorphic, then $ a$ is an essential singularity of $ f$ if and only if the image of any punctured neighborhood of $ a$ under $ f$ is dense in $ \mathbb{C}$. Put another way, a holomorphic function can come in an arbitrarily small neighborhood of its essential singularity arbitrarily close to any complex value.



"Casorati-Weierstrass theorem" is owned by PrimeFan. [ full author list (2) | owner history (1) ]
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See Also: Picard's theorem

Other names:  Weierstrass-Casorati theorem

Attachments:
proof of Casorati-Weierstrass theorem (Proof) by pbruin
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Cross-references: complex, dense in, neighborhood, image, essential singularity, holomorphic, domain
There are 4 references to this entry.

This is version 4 of Casorati-Weierstrass theorem, born on 2003-04-03, modified 2008-06-18.
Object id is 4143, canonical name is CasoratiWeierstrassTheorem.
Accessed 5362 times total.

Classification:
AMS MSC: 30D30 (Functions of a complex variable :: Entire and meromorphic functions, and related topics :: Meromorphic functions, general theory)
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