pole is well defined on the Riemann surface

Question

I want to show a pole is well defined on the Riemann surface.

Let $M$ is a Reimann surface, $f:M \rightarrow \hat{C}$ and $f(p)=\infty$. Suppose $(U,\varphi) $ and $(V, \psi)$ are two charts of M and $p \in V \cap U$. $f o \varphi^{-1}$ has a pole of orde $t$ at $\varphi^{-1}(p)$ and $f o \psi^{-1}$ has a pole of orde $l$ at $\psi^{-1}(p)$.

How can we prove $t=l$?

Answer 1

Both maps differ by a holomorphic map $\varphi \circ \psi^{-1}$ and the order of a zero (or pole) does not hange under a holomorphic coordinate change.