Let $U \subset \mathbb{C}$ be open, let $a \in U$, let $m \in \mathbb{N}$, and let $f: U \setminus \{a\} \to \mathbb{C}$ be holomorphic. Then $a$ is a pole of order $m$ of $f$ if and only if there exists $r > 0$ such that $B(a,r) \subset U$, $f(z) \ne 0$ for every $z \in B(a,r) \setminus \{a\}$, and the reciprocal map
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\begin{align*}
z \mapsto \frac{1}{f(z)}
\end{align*}
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on $B(a,r) \setminus \{a\}$ extends to a [holomorphic function](/page/Holomorphic%20Function) $F: B(a,r) \to \mathbb{C}$ having a zero of order $m$ at $a$.