Suppose $p\colon Y \rightarrow X$ is a holomorphic mapping of Riemann surfaces, $a\in X$, $b\in p^{-1}(a)$ and $k$ is the multiplicity of $p$ at $b$.
Then I am trying to show the claim:
Given any holomorphic $1$-form $\omega$ on $X\setminus\{a\}$ we have $$\operatorname{Res}_b(p^*\omega)= k \operatorname{Res}_a(\omega).$$
The residue of $\omega$ at $a$ is defined to be the coefficient of the exponent $z^{-1}$ in the Laurent expansion (with respect to a chart $z$ of $X$ centered at $a$) of the coefficient function $f$ of $\omega$ with respect to the chart $z$ (i.e. $\omega=fdz$). The residue of $p^*\omega$ at $b$ is defined similarly.
I tried to somehow use the local normal form of $p$ to show above claim, but I can't quite figure out how. Any hints?
Residue at a point $\alpha$ is the integral over a circle around point $\alpha$ $$\dfrac{1}{2\pi i}\int_{\gamma} \omega$$ Now, since every holomorphic map locally looks like a power map, a small circle around $\alpha$ maps to a circle wound mult$_{\alpha}(f)$ times . ($e^{i2\pi \cdot t}\mapsto e^{i2\pi\cdot k\cdot t}$) \begin{align*} \text{res}_{a}(f^* \omega) &= \dfrac{1}{2\pi i} \int_{\gamma} f^* \omega \\ &= \dfrac{1}{2\pi i} \int_{f(\gamma)} \omega \\ &= \text{mult}_{a}(f)\cdot \text{res}_{f(a)}(\omega) \end{align*}