I've attempted to solve the following textbook problem, but now I'm stuck.
Find a Möbius transformation which maps the disc $|z-2|<2$ onto the unit disc $|z|<1$, maps the point 0 to the point 1, and maps the point 1 to $\frac{1}{2}i$.
The best result I've obtained this far is by the maps $z\mapsto z-2$, $z\mapsto\frac{z}{2}$, and $T_{\frac{1}{2}i}(z)=\frac{z-\frac{1}{2}i}{-\frac{1}{2}iz-1}$.
I've successfully obtained the unit circle and swapped 0 (which was 1 before) and $\frac{1}{2}i$. I don't understand how to map -1 (which was 0 before) to 1 without rotating $\frac{1}{2}i$ to another point.
As I use $T_{\frac{1}{2}i}(z)$ to map 0 to $\frac{1}{2}i$ the point -1 is mapped to $z=1+\frac{4}{3}i$ of which $|z|=\frac{5}{3}$ (i.e. outside $|z|=1$. I feel as I've hit a dead end. Any advice would be appreciated.
Do this transformation in three steps.
First map the disc $|z-2|<2$ to the unit disc by a translation and scaling: $w_1 = (z-2)/2$, taking $z = 0$ to $w_1 = -1$ and $z=1$ to $w_1 = -1/2$. The next Möbius transformations will preserve the unit disc: the Möbius transformations that do this are of the form
$$ z \mapsto \phi \dfrac{z - a}{1 - \overline{a} z}$$
where $|a| < 1$ and $|\phi| = 1$. It's convenient to first map $w_1 = -1/2$ to $w_2 = 0$: this will be the case if $a = -1/2$.
$$ w_2 = \phi \dfrac{w_1 + 1/2}{1 - w_1/2} $$
mapping $w_1=-1$ to $w_2 =- \phi$.
Similarly, a transformation $$z \mapsto \dfrac{z + i/2}{1 + i z/2}$$ preserves the unit circle and takes $0$ to $i/2$.
Now you just need to figure out what $\phi$ should be so that this last transformation maps $-\phi$ to $1$, and then take the composition of the three transformations.