Linear fractional transformation

The alternate name for a Möbius transformation, $w=\dfrac{az+b}{cz+d}$ .

Linear fractional transformation is the name used, interchangeably with Möbius transformation, for a map $w=\dfrac{az+b}{cz+d}$ with $ad-bc\neq 0$ . The name emphasizes that the map is a ratio (fraction) of two linear expressions in $z$ , in contrast to a plain complex linear map, which has no denominator at all.

Examples

Both names refer to the same map: $w=\dfrac{2z+1}{z-3}$ is called a Möbius transformation in some sources and a linear fractional transformation in others.