Definition:Möbius Transformation

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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:


A Möbius transformation is a mapping $f: \R^* \to \R^*$ of the form:

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

where:

$\R^*$ denotes the Alexandroff extension of the real number line
$a, b, c, d \in \R$ such that $a d - b c \ne 0$


We define:

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

if $c \ne 0$, and:

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


Also defined as

Some sources, when defining a Möbius transformation, do not insist that $a d - b c \ne 0$.

However, it needs to be pointed out that when $a d - b c = 0$, the resulting mapping is not a bijection.


Also known as

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


Also see

  • Results about Möbius transformations can be found here.


Do not confuse this with the Definition:Möbius Function.


Source of Name

This entry was named for August Ferdinand Möbius.


Sources