Proof That Extended Mobius Transformations Preserve Regions and Boundaries

Theorem

Letbe an extended mobius transformation and letbe a region in the domain of(not including the point infinity). Thenis a region and hat f maps the boundary of to the boundary ofin

Proof

is analytic and non constant onso it follows from the Open Mapping Theorem thatis a region.

To show thatmaps the boundary inofonto the boundary inofwe show that ifis any open disc incentred atwherethen there is an open disccentred atsuch thatThere are three cases:

a) Ifis in the domain ofandthenis continuous atso for each discthere is a discsuch that

b) Ifandis a pole ofthenasso for each disccentred atthere issuch thatwhenever

MoreoversowheneverThus ifthen

c) Ifthenhas a removable singularity or pole atso for a disccentred atthere issuch thatwheneverbut this means thatwheneverand so ifis the discthen

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 bothandIt follows thatand hencecontains points from bothandsois a boundary point inof

To show the mapping is onto we apply the same argument toand show that ifis a boundary point inofthenis a boundary point inofThe result follows.