Find a Mobius transformation that maps the unit-disk onto unit-disk: $1\to1$ and $1+i\to \infty$

Question

I tried to find the Moebius transformation which maps unit-disk to unit-disk and maps 2 points: $z_1=1$ to $w_1=1$ and $z_2=1+i$ to $w_2=\infty$.

What I have:

1. $w_1= \dfrac{a+b}{c+d} = 1$
2. $w_2= \dfrac{a(1+i)+b}{c(1+i)+d} = \infty$

then $c(1+i)+d=0$ and $d = -c(1+i)$

What is the next step? It is not enough equations to find all a,b,c,d.

Answer 1

Every Möbius transformation of the type$$z\mapsto\omega\frac{z-a}{\overline az-1},$$with $|\omega|=1$, maps the unit disk into itself. If you want it to map $1+i$ into $\infty$, take $a$ such that $\overline a(1+i)-1=0$; in other words, take $a=\frac12+\frac i2$. So, $f(1)=\omega i$. Therefore, take $\omega=-i$, and your Möbius transformation will be$$z\mapsto\frac{(-1+i)z+1}{-z+1+i}.$$