By considering the singularity at $\infty$, show that any injective analytic map $f: \mathbb{C} \rightarrow \mathbb{C}$ has the form $f(z)=a z+b$ for some $a \in \mathbb{C}^{*}$ and $b \in \mathbb{C}$.

State the Riemann-Hurwitz formula for a non-constant analytic map $f: R \rightarrow S$ of compact Riemann surfaces $R$ and $S$, explaining each term that appears.

Suppose $f: \mathbb{C}_{\infty} \rightarrow \mathbb{C}_{\infty}$ is analytic 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 $z \mapsto z^{2}$.