Is there a Möbius transformation mapping the upper half plane onto itself that interchange two preassigned points in the upper half plane? If so , how many such Möbius transformations are there?
Here are steps that I followed:
1.we know we can transform upper half plane onto unit circle.
- I want to find a Möbius transformation that maps unit circle to unit circle but interchanging the preassigned points.
3. Finally , if I find a transformation that maps the unit circle to the upper half plane.Then I think we are done.
I have problem in finding the Möbius transformation that maps unit circle to unit circle with interchanging property that already mention.
Does someone have any idea about this?