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
Canonical name	AutomorphismsOfUnitDisk
Date of creation	2013-03-22 13:36:48
Last modified on	2013-03-22 13:36:48
Owner	brianbirgen (2180)
Last modified by	brianbirgen (2180)
Numerical id	6
Author	brianbirgen (2180)
Entry type	Example
Classification	msc 30C20
Related topic	MobiusTransformation
Related topic	ProofOfConformalMobiusCircleMapTheorem