Further Analysis | Part IB, 2003

(a) Let f:[1,)Cf:[1, \infty) \rightarrow \mathbb{C} be defined by f(t)=t1e2πitf(t)=t^{-1} e^{2 \pi i t} and let XX be the image of ff. Prove that X{0}X \cup\{0\} is compact and path-connected. [Hint: you may find it helpful to set s=t1.]\left.s=t^{-1} .\right]

(b) Let g:[1,)Cg:[1, \infty) \rightarrow \mathbb{C} be defined by g(t)=(1+t1)e2πitg(t)=\left(1+t^{-1}\right) e^{2 \pi i t}, let YY be the image of gg and let Dˉ\bar{D} be the closed unit disc{zC:z1}\operatorname{disc}\{z \in \mathbb{C}:|z| \leq 1\}. Prove that YDˉY \cup \bar{D} is connected. Explain briefly why it is not path-connected.

Typos? Please submit corrections to this page on GitHub.