1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Möbius transformations.

  1. Jun 5, 2005 #1
    Could someone please explain me what are Möbius transformations, and what do they work for?
    Where can I find more info about this?
    Thanks in advance.
  2. jcsd
  3. Jun 5, 2005 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    They are the set of all functions of the form

    [tex]f(z)= \frac{az+b}{cz+d}[math]

    subject to the condition the ad-bc=/=0

    They are the set of all "rational bijections" of the extended complex plane (ie allowing a point at infinity) that preserves the set of circles and straight lines.
    (ie they are the automorphism group of the riemann sphere)
    Rational means is of the form P(z)/Q(z) where P and Q are polynomials in z.

    There is plenty of information out there (google) and I believe there is a book by Alan Beardon about geometry and analysis that will explain this stuff.

    The important thing is they are a group of meromorphic functions on the extended plane and we can describe lots of geometry in terms of them.

    The set of transformations with ad-bc=1is isomorphic with SU(2,C).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook