I am trying to solve the following problem:
Let $f$ be a meromorphic function on a Riemann surface $X$, and let $f(p)=\infty$. Show that for any chart $(U_{\alpha},\phi_{\alpha})$ with $p\in U_{\alpha}$ the function $f\circ \phi_{\alpha}^{-1}$ is meromorphic in $\phi_{\alpha} (U_{\alpha})$ and has a pole at $\phi_{\alpha}(p)$.
Furthermore, the order of this pole is independent of the chart and the residue is not necessarily independent of the chart.
Now I am working on the second part. Here is my attempt:
Pick up another chart $(\tilde{U}_{\alpha},\tilde{\phi}_{\alpha})$, then $$f\circ \tilde{\phi}_{\alpha}^{-1}=f\circ \phi_{\alpha}^{-1}\circ \phi_{\alpha}\circ \tilde{\phi}_{\alpha}^{-1}=(f\circ \phi_{\alpha}^{-1})\circ( \phi_{\alpha}\circ \tilde{\phi}_{\alpha}^{-1})$$ So it is enough to prove the following result:
For a biholomorphic function $h$ and a meromorphic function $g$, we have $$\mathrm{ord}_p g=\mathrm{ord}_p (g\circ h).$$
Suppose $$g(z)=\frac{a(z)}{(z-p)^k}$$ for some holomorphic function $a(z)$. Then $$g(h(z))=\frac{a(h(z))}{((z-p)+o(z-p)-p)^k}=\frac{1}{(z-p)^k}h'$$ for some holomorphic function $h'$.