Definition:Möbius Transformation/Real Numbers
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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 $2$