4.I.4E

(i) Let $D$ be the open unit disc of radius 1 about the point $3+3 i$. Prove that there is an analytic function $f: D \rightarrow \mathbb{C}$ such that $f(z)^{2}=z$ for every $z \in D$.

(ii) Let $D^{\prime}=\mathbb{C} \backslash\{z \in \mathbb{C}: \operatorname{Im} z=0$, Re $z \leqslant 0\}$. Explain briefly why there is at most one extension of $f$ to a function that is analytic on $D^{\prime}$.

(iii) Deduce that $f$ cannot be extended to an analytic function on $\mathbb{C} \backslash\{0\}$.

*Typos? Please submit corrections to this page on GitHub.*