Let $\alpha$ be a holomorphic 1-form on a Riemann surface $X$ s.t. \begin{align*} \alpha \vert _p=0 \\ f dz=(u+iv) dz \textit{ on U} \end{align*}$(U, \varphi)=(U, z)$ ia a coordinate disk centered at $p$ such that $\alpha$ does not vanish on $U-p$.
Then $f$ is a holomorphic function thus has an integer multiplicity, say $n$. $u,-v:\varphi^{-1}(S^1)\to \mathbb{R}^2 -0 $ is a closed loop and thus has an integer winding number around $0$, say $m$.
How can I show that $n=m$?