(i) Show that transformations of the complex plane of the form

$$\zeta=\frac{a z+b}{c z+d}$$

always map circles and lines to circles and lines, where $a, b, c$ and $d$ are complex numbers such that $a d-b c \neq 0$.

(ii) Show that the transformation

$$\zeta=\frac{z-\alpha}{\bar{\alpha} z-1}, \quad|\alpha|<1$$

maps the unit disk centered at $z=0$ onto itself.

(iii) Deduce a conformal transformation that maps the non-concentric annular domain $\Omega=\{|z|<1,|z-c|>c\}, 0<c<1 / 2$, to a concentric annular domain.