Suppose $M$ is a Riemann surface and $f$ is a meromorphic function on $M$ who has a pole at $p$ of degree $n$. If $w,z$ are both local coordinates around $p$ such that $w(0)=z(0)=p$, then $f$ has a Laurent series power at $p$ as $$ f(z) =\sum_{j=-n}^\infty a_{j}z^j $$ $$ f(w) =\sum_{j=-n}^\infty b_{j}w^j $$
will $a_{-n}=b_{-n}$ holds?