= Riemann-Hurwitz formula
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{wiki=Riemann–Hurwitz_formula}
For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum(e_p-1)$.
Let $f:X\to Y$ be a nonconstant <holomorphic map> between <compact Riemann surfaces> that are <connected spaces>, without boundary. Write $g_X,g_Y$ for their <genera> and $d\geq1$ for the <degree of a holomorphic map>. With <analytic ramification index> $e_p$ at each $p\in X$, the formula is
$$
2g_X-2=d(2g_Y-2)+\sum_{p\in X}(e_p-1),\qquad
\chi(X)=d\chi(Y)-\sum_{p\in X}(e_p-1).
$$
Here $\chi=2-2g$ is the <Euler characteristic>. Only finitely many <ramification points> contribute. The <holomorphic map> is automatically <surjective>; the <valency theorem> gives $\sum_{p\in f^{-1}(y)}e_p=d$ even at a <branch value of a holomorphic map>. Constant maps are excluded.<Riemann Surfaces by Constantin Teleman (Lent Term 2003)>[Lecture 5, Theorems 5.1 and 5.3; Definition 5.16 and Theorem 5.17, printed pp. 16-19]{ref}
The useful interpretation is that <total ramification of a holomorphic map> measures how much the source <Euler characteristic> falls short of $d$ copies of the target. The <ramification of a power map of the Riemann sphere> spends that entire correction at two points; the <double cover of the sphere with four branch values> spends it at four points and produces a <genus>-one surface.
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