Riemann-Hurwitz formula

Statement

Suppose $M, N$ are compact Riemann surfaces and $f : M \to N$ is an analytic map. Then we have:

$g_{M} - 1 = d e g (f) (g_{N} - 1) + \frac{1}{2} B$

where $B$ , termed the branching number of $f$ , is defined as:

$B : = \sum_{p \in M} (v_{f} (p) - 1)$

where $v_{f} (p)$ is the local degree of the map $f$ around $p$ . Note that the set of $p$ for which $v_{f} (p) \neq 1$ is a finite set, because it is discrete and closed and $M$ is a compact set. The points $p$ for which $v_{f} (p) \neq 1$ are termed branch points.

Note that in case $B = 0$ , we get an actual covering map, and in this case we are simply told that the degree of the covering map equals the ratio of the Euler characteristics of the Riemann surfaces.