Fixed Points of an Extended Mobius Transformation

Ifis an extended Mobius transformation (with domain and image both including the point at infinity in the complex plane) then the fixed points ofare the solutions to


Ifthenis an extended linear function which hasas one of its fixed points. If also,then there is one other fixed point which is the solution to

Ifthensois not a fixed point and the fixed point(s) ofmust lie inThe fixed points will can be found from the equation

We can rearrange this to give

This is a quadratic equation, which must have one or two solutions in the complex plane.

Example: Find the fixed points of the extended Mobius transformation

is not a fixed point sinceWe solve

The solutions to this equation areand

This Mobius transformation has two fixed pointsand