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Math Help - Möbius transformation and fixed points - complex analysis

  1. #1
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    Question Möbius transformation and fixed points - complex analysis

    Let f:z\mapsto w = {az+b \over cz+d} \ \ \ (ad-bc \ne 0) be a Möbius transformation other than the identity map. A point \alpha in \mathbb{C} is said to be a fixed point of f if f(\alpha ) = \alpha.

    Suppose that f has distinct fixed points, \alpha and \beta. Prove that {w-\alpha \over w-\beta}=k{z-\alpha \over z-\beta}, where k={a-c\alpha \over a-c\beta}.

    What is the image under f of
    (i) the circline |{z-\alpha \over z-\beta}|=\lambda,
    (ii) the arc \arg((z-\alpha )(z-\beta )) = \mu \pmod{2\pi}?
    Last edited by JJMC89; September 12th 2010 at 10:41 AM.
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  2. #2
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    Anyone have any ideas?
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