(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, $\operatorname{Re}(z)>0$, to the unit disc

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

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