# Thread: Möbius transformation and fixed points - complex analysis

1. ## Möbius transformation and fixed points - complex analysis

Let $f:z\mapsto w = {az+b \over cz+d} \ \ \ (ad-bc \ne 0)$ be a Möbius transformation other than the identity map. A point $\alpha$ in $\mathbb{C}$ is said to be a fixed point of $f$ if $f(\alpha ) = \alpha$.

Suppose that $f$ has distinct fixed points, $\alpha$ and $\beta$. Prove that ${w-\alpha \over w-\beta}=k{z-\alpha \over z-\beta}$, where $k={a-c\alpha \over a-c\beta}$.

What is the image under $f$ of
(i) the circline $|{z-\alpha \over z-\beta}|=\lambda$,
(ii) the arc $\arg((z-\alpha )(z-\beta )) = \mu \pmod{2\pi}$?

2. Anyone have any ideas?