Q:

Möbius transformation question

Möbius transformation copies the annulus $\{z:r<|z|<1\}$ to the domain between $\{z:|z-1/4|=1/4\}$ and $\{z:|z|=1\}$
Please help me to find what is $r$.

A:

Hint: The inverse of this transform is also Möbius and might for simplicity map $1\mapsto 1$, $-1\mapsto-1$, $\frac12\mapsto r$, $0\mapsto -r$. Can you find $r$ in this special case? Can you justify the simplification?

