Past exam of the geometry course of the University of Oxford / 2007 / B / 7 / ii / Solution

For $d>1$, the map $z\mapsto z^d$ has local index $d$ at zero and at infinity; at infinity use coordinates $w=1/z$ in source and target, giving again $w\mapsto w^d$. At every nonzero finite point its derivative is nonzero, so there are no further ramification points. Thus the ramification sum is $2(d-1)$ and the formula gives $-2=-2d+2(d-1)$, as required for the sphere. For $d=1$ the map is the identity and has no ramification, giving the same equality separately.