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essential singularity (Definition)

Let $ U\subset\mathbb{C}$ be a domain, $ a\in U$, and let $ f:U \setminus \{a\} \to \mathbb{C}$ be holomorphic. If the Laurent series expansion of $ f(z)$ around $ a$ contains infinitely many terms with negative powers of $ z-a$, then $ a$ is said to be an essential singularity of $ f$. Any singularity of $ f$ is a removable singularity, a pole or an essential singularity.

If $ a$ is an essential singularity of $ f$, then the image of any punctured neighborhood of $ a$ under $ f$ is dense in $ \mathbb{C}$ (the Casorati-Weierstrass theorem). In fact, an even stronger statement is true: according to Picard's theorem, the image of any punctured neighborhood of $ a$ is $ \mathbb{C}$, with the possible exception of a single point.



"essential singularity" is owned by pbruin.
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See Also: Laurent series, pole, removable singularity, Picard's theorem, Riemann's removable singularity theorem
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Cross-references: point, Picard's theorem, even, Casorati-Weierstrass theorem, dense in, neighborhood, image, pole, removable singularity, powers, negative, terms, contains, Laurent series, holomorphic, domain
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This is version 4 of essential singularity, born on 2003-03-28, modified 2003-04-03.
Object id is 4132, canonical name is EssentialSingularity.
Accessed 4120 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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