I was trying to solve this problem but had problems:
Find a conformal map from $D:=\{z∈C:|z|<1 \quad \text{and}\quad |z-(1+i)|<1\}$ the unit disc in $\mathbb{C}$.
I want to map $D$ to the upper half plane, so I can compose it with the Cayley transformation so I considered $$f(z)=\frac{z-i}{z-1}$$ which has $f(i)=0$ and $f(1)=\infty$. This Moebius transformation should supposedly map to a quadrant(though im not sure how to conclude which). After determining the quadrant, I need to find a value $\beta$ such that $f(z)^{\beta}$ maps to the upper half plane. How do I determine these things?
Your f(z) being a mobius transfromation maps cicrlines to circlines. So, it maps the circles |z|=1 and |z-(1+i)|=1 to lines through the origin as their images contain o and infinity. The question remains, which lines? Evaluate any other point on each of the circles. An easy candidate will be the appropriate scalar multiples of 1+i. By appropriate I mean |c(1+i)|=1 and |d(1+i)-(1+i)|=1. Once you know, evaluate these in f to figure out which pair of lines you have. Now, just pick a point in D and evaluate it in f to find which of the four sectors cut out D will map too.
Then, if necessary, apply a rotation so one line lies on the positive reals. Then, raise to the appropriate exponent so that the other line ends up on the negative reals, giving you D mapping to the upper half plane. The exponent is found by finding the angle between the lines. The exponent times this angle (in radians) should equal pi.
Hope this helps.