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!