Mapping the upper half plane to unit disc

Question

Find a Mobius transformation from the closed upper half plane onto the closed unit disc taking $1 + i$ to $0$ and $1$ to $−i$.

So far I have the Cayley map: $M(z)=\frac{z-i}{z+i}$ maps the upper half plane to the unit circle,

I also have a mapping from the unit circle to unit disk as $$f(z) = e^{i\theta}\frac{z - \beta}{1 - {\beta}z}.$$

I then thought of doing $M(z)\circ f(z)$ however when I input the values $1 + i$ and 1 they do not get the required values $0$ and $-i$.

Where have I gone wrong? Thanks!

Answer 1

Hint. You should consider the composition $f(M(z))$ with $$M(z)=\frac{z-i}{z+i}\quad\mbox{and}\quad f(z) = e^{i\theta}\frac{z - \beta}{1 - \overline{\beta}z}.$$ We have that $\beta:=M(1+i)=1/(1+2i)$ (note that $|\beta|<1$).

Finally use $f(M(1))=f((1-i)/(1+i))=-i$ to find $e^{i\theta}$. It turns out that $e^{i\theta}=(4+3i)/5$.