Let $U \subset \mathbb{C}$ be open, let $a \in U$, and let $p: U \to \mathbb{C}$ and $q: U \to \mathbb{C}$ be holomorphic functions. Suppose that $q$ has a zero of order $s \in \mathbb{N}$ at $a$, with $s \geq 1$. Let $r$ be an integer with $0 \leq r < s$, and suppose that either $r = 0$ and $p(a) \neq 0$, or $r \geq 1$ and $p$ has a zero of order $r$ at $a$.

Define the zero set

\begin{align*} Z(q) := \{z \in U : q(z) = 0\} \end{align*}

and define

\begin{align*} f: U \setminus Z(q) \to \mathbb{C} \end{align*}

by

\begin{align*} f(z) := \frac{p(z)}{q(z)}. \end{align*}

Then $f$ has a pole of order $s-r$ at $a$.