pole is well defined on the Riemann surface 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 $t=l$?

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.
