There is a lemma in Rick Miranda's Book "Algebraic Curves and Riemann surfaces " is the following,
Let $f$ be a meromorphic function on a Riemann surface $X$ , with associated holomorphic map $F:X\rightarrow \mathbb{C}_{\infty} $ , show that , If $p$ is neither a zero nor a pole of $f$, then $mult_p(F)=ord_p(f-f(p))$.
What is holomorphic map associated to a meromorphic function?