Category:Definitions/Möbius Transformations

This category contains definitions related to Möbius Transformations.
Related results can be found in Category:Möbius Transformations.

A Möbius transformation is a mapping $f: \overline \C \to \overline \C$ of the form:

$\map f z = \dfrac {a z + b} {c z + d}$

where:

$\overline \C$ denotes the extended complex plane

$a, b, c, d \in \C$ such that $a d - b c \ne 0$

We define:

$\map f {-\dfrac d c} = \infty$

if $c \ne 0$, and:

$\map f \infty = \begin {cases} \dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end {cases}$