2003-21.md

course	course_year	question_number	tags	title	year
Further Analysis	IB	21	IB 2003 Further Analysis	2.II.13E	2003

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

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