\begin{definition}[Definition:Möbius Transformation]
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:
:$\map f \infty = \begin{cases} \dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end{cases}$
\end{definition}