Further Analysis | Part IB, 2004

(i) Let DD be the open unit disc of radius 1 about the point 3+3i3+3 i. Prove that there is an analytic function f:DCf: D \rightarrow \mathbb{C} such that f(z)2=zf(z)^{2}=z for every zDz \in D.

(ii) Let D=C\{zC:Imz=0D^{\prime}=\mathbb{C} \backslash\{z \in \mathbb{C}: \operatorname{Im} z=0, Re z0}z \leqslant 0\}. Explain briefly why there is at most one extension of ff to a function that is analytic on DD^{\prime}.

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

Typos? Please submit corrections to this page on GitHub.