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, $$ f(z) = \sum_{k=-n}^\infty c_k (z-a)^k $$ 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.