In Griffiths' Introduction to Algebraic Curves. In the proof of the following statement,
Let $\omega$ be a meromorphic 1-form on a compact Riemann surface $C$, then $\sum_{p\in C} \operatorname{mult} _p(\omega) =-\chi(C) $.
The author asserted that we can find a suitable coordinate to let $\omega$ have the expression $z^v dz$ where $v$ is the multiplicity of zero or pole of $w$. But I can only do it for meromorphic functions instead of meromorphic differentials. How can I work this out?