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Math Help - Mobius transformations

  1. #1
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    Mobius transformations

    I need help to find the family of Fractional-Linear (Mobius) transformations that map the unit circle onto itself
    a tranformation includes a translation, inversion, magnification, rotation and translation. and takes the form w(z)=(az+b)/(cz+d)
    Whenever i work through these i get {a,b,c,d}=1 so this isn't helpful to me. any help would be great
    cheers
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  2. #2
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    Quote Originally Posted by i_never_noticed View Post
    I need help to find the family of Fractional-Linear (Mobius) transformations that map the unit circle onto itself
    a tranformation includes a translation, inversion, magnification, rotation and translation. and takes the form w(z)=(az+b)/(cz+d)
    Whenever i work through these i get {a,b,c,d}=1 so this isn't helpful to me. any help would be great
    cheers
    Let f be a Mobius trasformation, so f(z) = \tfrac{az+b}{cz+d} with ad-bc\not = 0. There are two cases to consider. One case is easy, the other case is more difficult. The first case is when c=0, when this happens we just have f(z) = Az+B. Thus, f is just a translation, rotation, and magnification. Under these conditions the only way to map a unit circle to a unit circle is when the magnification factor is 1, and the translation is zero. However, you can rotate the circle as much as you like, therefore f(z) = e^{i\theta} z.

    In the second case f(z) = \tfrac{az+b}{cz+d},c\not =0 the Mobius transformation will always have a singularity (at z=-b/c). The singularity cannot lie on the circle because otherwise the circle will get mapped into a line. The singularity either lies inside the circle or outside the circle. If it lies outside the circle then the f maps the inside of the disk onto itself and the boundary of the disk onto itself by the maximum modolus theorem. It can now be shown that f must have the form f(z) = e^{i\theta}\cdot \frac{z-\alpha}{1-\bar \alpha z} where |\alpha|<1 and conversely. So this is the complete description of Mobius transformations which have their singularity outside the circle. I did not solve the problem yet of what happens if the singularity lies inside of the circle, I am thinking of considering the function 1/f but so far I was unable to complete the argument in the second case.
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