Consider a function $f:\mathbb{P}^1\to\mathbb{C}$ s.t. $f|_U(z)=z$ on $U=\{[x:y]\in \mathbb{P}^1|x\neq0\}$ where $z=\frac{y}{x}$ and $f|_V(w)=w$ on $V=\{[x:y]\in \mathbb{P}^1|y\neq0\}$ where $w=\frac{x}{y}$. I want to make sure that I understand the definition of a meromorphic function.
We say that $f$ is meromorphic if, on each open $U_i$, $f$ can be expressed as a ratio of two holomorphic functions $f_i$, $g_i$, i.e. $f|_{U_i}=\frac{f_i}{g_i}$, and $f_ig_j=f_jg_i$ on $U_i\cap U_j$.
So, in our case, we have two open sets $U$ and $V$ with the local expressions $f|_U(z)=\frac{z}{1}$ and $f|_V(w)=\frac{w}{1}$. We want to make sure that $f|_U$ and $f|_V$ coincide on the intersection $U\cap V$. But, this follows from the fact that $g_1=g_2=1$, and $f_1$ with $f_2$ are identity maps i.e. $$f_i(t)g_j(t)=f_j(t)g_i(t)\Rightarrow t\cdot 1=t\cdot 1,\forall t\in U\cap V.$$ Does it make sense ?
Using the same approach, we can show, for example, that $f(z)=\frac{z^3}{z^2-1}$ is a meromorphic function.