# automorphisms of unit disk

All automorphisms of the complex unit disk $\Delta=\{z\in\mathbb{C}:|z|<1\}$ to itself, can be written in the form $f_{a}(z)=e^{i\theta}\frac{z-a}{1-\overline{a}z}$ where $a\in\Delta$ and $\theta\in S^{1}$.

This map sends $a$ to $0$, $1/\overline{a}$ to $\infty$ and the unit circle to the unit circle.

Title automorphisms of unit disk AutomorphismsOfUnitDisk 2013-03-22 13:36:48 2013-03-22 13:36:48 brianbirgen (2180) brianbirgen (2180) 6 brianbirgen (2180) Example msc 30C20 MobiusTransformation ProofOfConformalMobiusCircleMapTheorem