Ifis an extended Mobius transformation (with domain and image both including the point at infinity in the complex plane) then the fixed points of
are the solutions to
or
Ifthen
is an extended linear function which has
as one of its fixed points. If also,
then there is one other fixed point which is the solution to
Ifthen
so
is not a fixed point and the fixed point(s) of
must lie in
The 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 since
We solve
The solutions to this equation areand
This Mobius transformation has two fixed pointsand