Define what is meant by a Mobius transformation. Determine the unique ¨ Mobius transformation which sends
- $z = −1$ to $w = i$,
- $z =\infty$ to $w = 1$,
- $z = i$ to $w = 1 + i$.
Verify that, for this Mobius transformation, the real axis in ¨ the $z$-plane is mapped onto the circle of radius $1$ centered at the origin in the $w$-plane.
When I was working this out I got the Mobius transformation to be $$z\mapsto {(2-i)z+1\over 1-iz}$$
This worked for mapping $-1$ and infinity to the circle of radius $1$ however did not work for $1$ as it did not map to the circle radius $1$.