Proposition

Let $\gamma$ be the unit circle. Find a Mobius transformation that transforms $\gamma$ onto $\gamma$ and transforms $0$ to $1/2$.

Solution

We will use the formula given here.

$f(z) = (z - 1/2) / (1 - z/2)$ maps $\gamma$ onto $\gamma$ and sends 1/2 to 0. It suffices to take the inverse. Therefore, the answer is $g(z) = (2z - 1) / (-z + 2)$.