Integration of $\frac{1}{2\pi i}\frac{df}{f}$ over a closed curve is an integer

Question

I am trying to prove the following: $X$ be a compact Riemann Surface. For any smooth function $f\in\mathcal{E}(X)$ and $\alpha$ a closed curve $\frac{1}{2\pi i}\int_{\alpha}\frac{df}{f}$ is an integer. This is given as a hint in a problem in Forster's Riemann Surface book. So, enough to show that the integral is an integer for all the basis elements of $H_1(X,\mathbb{Z})$. My intuition is we should use that locally we have a branch of logarithm, but $f$ is only smooth. Can anyone please help me with a rigorous argument?

Answer 1

This was actually pretty easy. So, let's say $\gamma:[0,1]\rightarrow X$ be the closed curve. Then $f\circ\gamma$ becomes a closed curve in $\mathbb{C}.$ Then $\frac{1}{2\pi i}\int_\gamma\frac{df}{f}=\frac{1}{2\pi i}\int_{f\circ\gamma}\frac{dz}{z}.$ So, this is the winding number of the curve $f\circ\gamma$ around $0$. So, it's an integer.