Meromorphic function

Definition

Let $U \subseteq \mathbb{C}$ be an open set. A function $f$ is meromorphic on $U$ if there is a set $P \subseteq U$ with these properties.

1. $P$ has no accumulation point in $U$.
2. $f$ is holomorphic on $U \setminus P$.
3. Each point of $P$ is a pole of $f$.

Remarks

Rational functions $p/q$, with polynomials $p$ and $q \neq 0$, are meromorphic on $\mathbb{C}$. So are $\tan z$, $\frac{1}{\sin z}$ and the Gamma function. If $U$ is a domain, $g$ and $h$ are holomorphic on $U$, and $h$ is not identically zero, then $g/h$ is meromorphic on $U$. The meromorphic functions on a domain form a field. The function $e^{1/z}$ is not meromorphic on $\mathbb{C}$, because its singularity at $0$ is essential. The set $P$ can be infinite. It is countable, and each compact subset of $U$ contains only finitely many points of $P$.

Explanations

L1 human

A meromorphic function is holomorphic except at some isolated points, where it has poles. For example, $\frac{1}{z^2 + 1}$ is holomorphic on $\mathbb{C}$ except at $i$ and $-i$, where it goes to infinity.