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 $p$ has a zero of positive integer order $r$ at $a$ and that $q$ has a zero of positive integer order $s$ at $a$. If $r \ge s$, then there exists an open neighbourhood $W \subset U$ of $a$ such that $q(z) \ne 0$ for every $z \in W \setminus \{a\}$, and the quotient function $p/q: W \setminus \{a\} \to \mathbb{C}$ is holomorphic.

Moreover:

- if $r=s$, then $p/q$ has a removable singularity at $a$; - if $r>s$, then $p/q$ extends to a [holomorphic function](/page/Holomorphic%20Function) on $W$ with a zero of order $r-s$ at $a$.