Let $U \subset \mathbb{C}$ be open, let $a \in U$, let $m \in \mathbb{N}$ with $m \geq 1$, and let $f: U \setminus \{a\} \to \mathbb{C}$ be holomorphic. Then $a$ is a pole of exact order $m$ of $f$ if and only if there exist $r > 0$ and coefficients $(c_k)_{k \geq -m}$ in $\mathbb{C}$ such that $B(a,r) \subset U$, the [Laurent series](/page/Laurent%20Series)

\begin{align*} \sum_{k=-m}^{\infty} c_k (z-a)^k \end{align*}

converges to $f(z)$ for every $z \in B(a,r) \setminus \{a\}$, and $c_{-m} \neq 0$.

Equivalently, in the Laurent expansion of $f$ about $a$ on some punctured disk $B(a,r) \setminus \{a\}$, every coefficient of $(z-a)^k$ with $k < -m$ is zero and the coefficient of $(z-a)^{-m}$ is nonzero.