In Miranda's Algebraic Curves and Riemann Surfaces he defines the multiplicity of a non-constant holomorphic map $F\colon X \rightarrow Y$ between Riemann surfaces as the unique integer $m$ such that there are local coordinates $\phi$ and $\psi$, near $p$ and $F(p)$ respectively, with $\psi\circ F\circ\phi^{-1}$ having the form $z\mapsto z^m$.
On page 45 he then tries to give a way to determine the multiplicity of a non-constant holomorphic map without determining its local normal form.
He writes:
Take any local coordinates $z$ near $p$ and $w$ near $F(p)$; say that $p$ corresponds to $z_0$ and $F(p)$ to $w_0$. In terms of these coordinates, the map $F$ may be written as $w=h(z)$ where $h$ is holomorphic.
What does the expression $w=h(z)$ actually mean? It seems that the domains and codomains do not really match up (i.e. $z\colon U\rightarrow V\subseteq \mathbb{C}$ and $w\colon \tilde{U}\rightarrow \tilde{V}\subseteq \mathbb{C}$, where $U\subseteq X$ and $\tilde{U}\subseteq Y$ are open, and $h\colon ?\rightarrow ?$). Is $h=w\circ F \circ z^{-1}$? Why does Miranda use this (to my eye) careless and weird notation?