Möbius transformation

A map $w=\dfrac{az+b}{cz+d}$ with $ad-bc\neq 0$ , also called a linear fractional transformation.

A Möbius transformation is a map of the form $w=\dfrac{az+b}{cz+d}$ , where $a,b,c,d\in\mathbb{C}$ satisfy $ad-bc\neq 0$ (the condition keeping the map from collapsing to a constant). Also called a linear fractional transformation, it is built by composing a complex linear map with, at most, one inversion, and it is always a bijection of the extended complex plane to itself. Möbius transformations always send generalized circles to generalized circles and preserve the cross-ratio of any four points.

Examples

$w=\dfrac{z-1}{z+1}$ sends $z=1\mapsto 0$ , $z=-1\mapsto\infty$ , and $z=0\mapsto -1$ .
The identity map $w=z$ and every complex linear map $w=az+b$ are Möbius transformations with $c=0$ .