local representation of a holomorphic map

Question

Suppose $f:C\to C'$ is a holomorphic map between two Riemann surfaces $C$ and $C'$. How can we choose appropriate local coordinates $(U,z)$ of $C$, and $(V,w)$ of $C'$, such that $f$ can be represented locally as $w=z^d$. Thanks!

Answer 1

Let $(U, y)$, $(V, w)$ be an arbitrary local coordinates, then locally we have the factorization

$$w(y) = y^d r(y),$$

where $r(y)$ satisfies $r(0)\neq 0$. Then $r(y)^{1/d}$ locally in a smaller neighborhood $U' \subset U$. Then with $z = y r(y)^{1/d}$ then

$$w (z) = z^d.$$