Proof That Extended Mobius Transformations Preserve Regions and Boundaries
Theorem
Let
be an extended mobius transformation and let
be a region in the domain of(not including the point infinity). Then
is a region and hat f maps the boundary of
to the boundary of
in
Proof
is analytic and non constant onso it follows from the Open Mapping Theorem thatis a region.
To show thatmaps the boundary in
ofonto the boundary inofwe show that if
is any open disc incentred at
where
then there is an open disc
centred at
such that
There are three cases:
a) Ifis in the domain ofand
thenis continuous atso for each disc
there is a disc
such that
b) If
andis a pole ofthen
as
so for each disc
centred at
there is
such that
whenever
Moreover
so
whenever
Thus ifthen
c) If
thenhas a removable singularity or pole atso for a disccentred atthere issuch that
whenever
but this means thatwhenever
and so ifis the disc
then
Next suppose thatis a boundary point inofand letbe any open disc centred atthen there is an open disccentred atsuch thatButis a boundary point inofand socontains points from bothand
It follows that
and hencecontains points from bothand
so
is a boundary point inof
To show the mapping is onto we apply the same argument to
and show that ifis a boundary point inofthen
is a boundary point inof
The result follows.