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pole (Definition)

Let $ U \subset \mathbb{C}$ be a domain and let $ a \in \mathbb{C}$. A function $ f: U \longrightarrow \mathbb{C}$ has a pole at $ a$ if it can be represented by a Laurent series centered about $ a$ with only finitely many terms of negative exponent; that is,

$\displaystyle f(z) = \sum_{k=-n}^\infty c_k (z-a)^k $
in some nonempty deleted neighborhood of $ a$, with $ c_{-n} \neq 0$, for some $ n \in \mathbb{N}$. The number $ n$ is called the order of the pole.



"pole" is owned by djao.
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See Also: essential singularity
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Cross-references: order, number, deleted neighborhood, exponent, negative, terms, Laurent series, function, domain
There are 32 references to this entry.

This is version 3 of pole, born on 2002-01-04, modified 2004-12-01.
Object id is 1200, canonical name is Pole.
Accessed 5094 times total.

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