(a) Prove that $z \mapsto z^{4}$ as a map from the upper half-plane $\mathbb{H}$ to $\mathbb{C} \backslash\{0\}$ is a covering map which is not regular.

(b) Determine the set of singular points on the unit circle for

$$h(z)=\sum_{n=0}^{\infty}(-1)^{n}(2 n+1) z^{n}$$

(c) Suppose $f: \Delta \backslash\{0\} \rightarrow \Delta \backslash\{0\}$ is a holomorphic map where $\Delta$ is the unit disk. Prove that $f$ extends to a holomorphic map $\tilde{f}: \Delta \rightarrow \Delta$. If additionally $f$ is biholomorphic, prove that $\tilde{f}(0)=0$.

(d) Suppose that $g: \mathbb{C} \hookrightarrow R$ is a holomorphic injection with $R$ a compact Riemann surface. Prove that $R$ has genus 0 , stating carefully any theorems you use.