State the Classical Monodromy Theorem for analytic continuations in subdomains of the plane.

Let $n, r$ be positive integers with $r>1$ and set $h(z)=z^{n}-1$. By removing $n$ semi-infinite rays from $\mathbb{C}$, find a subdomain $U \subset \mathbb{C}$ on which an analytic function $h^{1 / r}$ may be defined, justifying this assertion. Describe briefly a gluing procedure which will produce the Riemann surface $R$ for the complete analytic function $h^{1 / r}$.

Let $Z$ denote the set of $n$th roots of unity and assume that the natural analytic covering map $\pi: R \rightarrow \mathbb{C} \backslash Z$ extends to an analytic map of Riemann surfaces $\tilde{\pi}: \tilde{R} \rightarrow \mathbb{C}_{\infty}$, where $\tilde{R}$ is a compactification of $R$ and $\mathbb{C}_{\infty}$ denotes the extended complex plane. Show that $\tilde{\pi}$ has precisely $n$ branch points if and only if $r$ divides $n$.