Definition:Möbius Transformation

Definition
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:


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

Real Numbers
The Möbius transformation is often seen restricted to the Alexandroff extension $\R^*$ of the real number line:

Also known as
Möbius transformations are also known as complex bilinear transformations or fractional linear transformations.

Also see

 * Möbius Transformation is Bijection
 * Möbius Transformations form Group under Composition

Do not confuse this with the Definition:Möbius Function.