(a) Define the degree $\operatorname{deg} f$ of a meromorphic function on the Riemann sphere $\mathbb{P}^{1}$. State the Riemann-Hurwitz theorem.

Let $f$ and $g$ be two rational functions on the sphere $\mathbb{P}^{1}$. Show that

$$\operatorname{deg}(f+g) \leqslant \operatorname{deg} f+\operatorname{deg} g$$

Deduce that

$$|\operatorname{deg} f-\operatorname{deg} g| \leqslant \operatorname{deg}(f+g) \leqslant \operatorname{deg} f+\operatorname{deg} g .$$

(b) Describe the topological type of the Riemann surface defined by the equation $w^{2}+2 w=z^{5}$ in $\mathbb{C}^{2}$. [You should analyse carefully the behaviour as $w$ and $z$ approach $\infty$.]