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Math Help - Find Möbius transformation

  1. #1
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    Find Möbius transformation

    Find the Möbius transformation that maps the points z=1,2,3 to w=i,-i,-1. What are its fixed points?

    By a Theorem, I know there's a transformation so that T(z_j)=w_j whenever z_1,z_2,z_3,w_1,w_2,w_3\in\mathbb C-\{0,1,\infty\}, so I took a generic transformation given by w(z)=\frac{az+b}{cz+d}, so I did i=\frac{a+b}{c+d}, -i=\frac{2a+b}{2c+d} and -1=\frac{3a+b}{3c+d}, then I think I need to find a relation, or something, how to solve this?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Find Möbius transformation

    Quote Originally Posted by Homing View Post
    i=\frac{a+b}{c+d}, -i=\frac{2a+b}{2c+d} and -1=\frac{3a+b}{3c+d},
    All right. Now write the corresponding linear system on the unknowns a,b,c,d , you'll find solutions of the form a=\alpha d,\;b=\beta d,\;c=\gamma d . So the transformation is w=\frac{\alpha d z+\beta d}{\gamma  dz+d}=\frac{\alpha  z+\beta }{\gamma z +1} .
    Last edited by FernandoRevilla; December 4th 2011 at 02:20 AM.
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    Re: Find Möbius transformation

    Okay so that includes that those constants could have the imaginary unit right?
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Find Möbius transformation

    Quote Originally Posted by Homing View Post
    Okay so that includes that those constants could have the imaginary unit right?
    Right. Also, after finding \alpha,\beta,\gamma you can easily check if the solution is correct, w(1)=\ldots=i etc.
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