2018-9.md

course	course_year	question_number	tags	title	year
Complex Analysis or Complex Methods	IB	9	IB 2018 Complex Analysis or Complex Methods	Paper 1, Section I, A	2018

(a) Show that

$$w=\log (z)$$

is a conformal mapping from the right half $z$-plane, $\operatorname{Re}(z)>0$, to the strip

$$S=\left{w:-\frac{\pi}{2}<\operatorname{Im}(w)<\frac{\pi}{2}\right}$$

for a suitably chosen branch of $\log (z)$ that you should specify.

(b) Show that

$$w=\frac{z-1}{z+1}$$

is a conformal mapping from the right half $z$-plane, $\operatorname{Re}(z)&gt;0$, to the unit disc

$$D={w:|w|<1}$$

(c) Deduce a conformal mapping from the strip $S$ to the disc $D$.