I know that the intersections of the two circles need to be sent to $0$ and $\infty$ in order to get a sector $S = \{x+iy: x>0,0<y<x\}$
The intersections of the two balls were i and -i. So i thought i would do the following:
$$i \rightarrow 0$$
$$-i \rightarrow \infty$$
$$-1 \rightarrow 1$$
this created the following mobiustransformation:
$$g(z) = \frac{z-i}{iz-1}.$$
the main problem is that i don't know where the other boundary-line of M is going. i tried filling in the value $z = 1-\sqrt{2}$, but i didn't succeed to find the correct solution. With this solution i can determine the angle of the sector and see which sector is the image of my function.
(The sector will have an scharp angle because the angles in the intersectionpoints are sharp, but what is the angle?)
Kees Til