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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

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    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).
     
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