Analytic mapping

Definition

Suppose $M,N$ are Riemann surfaces. A map $f:M\to N$ is termed an analytic map or analytic mapping if given any coordinate charts $\phi :U_1\to V_1\subset \mathbb{C}$ and $\psi :U_2\to V_2\subset \mathbb{C}$ on $M$ and $N$ respectively, the map:

$\psi \circ f\circ {\phi }^{-1}:U_1\cap f^{-1}(U_2)\to f(U_1\cap f^{-1}(U_2))\subset \mathbb{C}$

is a holomorphic function.