Definition:Möbius Transformation/Real Numbers

Definition

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 see

• Möbius Transformation is Bijection/Restriction to Reals
• Möbius Transformations form Group under Composition/Restriction to Reals

• 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

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 9$: Inverse Functions, Extensions, and Restrictions: Exercise $2$