2008-36.md

course	course_year	question_number	tags	title	year
Complex Methods	IB	36	IB 2008 Complex Methods	4.II.15C	2008

Let $H$ be the domain $\mathbb{C}-{x+i y: x \leq 0, y=0}$ (i.e., $\mathbb{C}$ cut along the negative $x$-axis). Show, by a suitable choice of branch, that the mapping

$$z \mapsto w=-i \log z$$

maps $H$ onto the strip $S={z=x+i y,-\pi<x<\pi}$.

How would a different choice of branch change the result?

Let $G$ be the domain ${z \in \mathbb{C}:|z|<1,|z+i|>\sqrt{2}}$. Find an analytic transformation that maps $G$ to $S$, where $S$ is the strip defined above.