where
and
satisfy
University Maths Notes: Complex Analysis – Mobius Transformations
A Mobius transformation is any rational function of the
formwhereandsatisfy
Properties:
Every Mobius transformation is conformal – preserves angles –
and analytic – differential on
Every Mobius transformation is either a linear function or a quotient of linear functions. If
then
and
Every Mobius transformation
may
be extended to
so
that ifthen
and
The
extended function is labelled
Extended
Mobius transformations map
one
to one onto
and
generalized circles – circles and lines – onto generalized
circles.
Ifthen
where
The set of extended Mobius transformations has the following group properties:
Closure: If
and
are
extended Mobius transformations then so is
Identity: The identity function in
is
an extended Mobius transformation.
Associativity: If
and
are
extended Mobius transformations then
An extended Mobius transformation
has
at least one fixed point
such
that
An extended Mobius transformation is defined by it's effect on three distinct points.
Given three distinct points
inand
any other three distinct points
in
there
is a unique extended Mobius transformation that sends
to
to
to
In
particular ifare
respectively
then
If
are
generalized circles then there is an extended Mobius transformation
that maps
to