Show that the analytic isomorphisms (i.e. conformal equivalences) of the Riemann sphere $\mathbb{C}_{\infty}$ to itself are given by the non-constant Möbius transformations.

State the Riemann-Hurwitz formula for a non-constant analytic map between compact Riemann surfaces, carefully explaining the terms which occur.

Suppose now that $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ is an analytic map of degree 2 ; show that there exist Möbius transformations $S$ and $T$ such that

$$S f T: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$$

is the map given by $z \mapsto z^{2}$.