polePole2002-01-04 01:42:522004-12-01 15:57:24Definition\usepackage{amssymb}
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\usepackage{xypic}Let $U \subset \mathbb{C}$ be a domain and let $a \in \mathbb{C}$. A function $f: U \longrightarrow \mathbb{C}$ has a {\em 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,
$$
f(z) = \sum_{k=-n}^\infty c_k (z-a)^k
$$
in some nonempty deleted neighborhood of $a$, with $c_{-n} \neq 0$, for some $n \in \mathbb{N}$. The number $n$ is called the \emph{order} of the pole.