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 see

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

Do not confuse this with the Definition:Möbius Function.