Looking at Definition 1.1.33 in the book Complex geometry by Huybrechts:
Def. Let $U \subset \mathbb{C}^n$ be open. A meromorphic function $f$ on $U$ is a function on the complement of a nowhere dense subset $S \subset U$ with the following property: there exist an open cover $U=\cup_i U_i$ and holomorphic functions $g_i,h_i:U_i\to\mathbb C$ with $h_i|_{U_i\setminus S}\cdot f|_{U_i\setminus S}=g_i|_{U_i\setminus S}$.
Isn't there a requirement on $h$ missing, i.e. $h_i \neq 0$ on $U_i\setminus S$?