Meromorphic function

Definition

On an open subset of complex numbers

Suppose $U$ is an open subset of $\mathbb{C}$. A meromorphic function on $U$ is defined in the following equivalent ways:

• A holomorphic function on $U\setminus D$, where $D$ is a discrete subset of $U$, such that all points of $D$ are poles for the function
• A holomorphic map from $U$ to the Riemann sphere, that is not the constant map to $\mathrm{\infty }$