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Thread: Find Möbius transformation

  1. #1
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    Find Möbius transformation

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

    By a Theorem, I know there's a transformation so that $\displaystyle T(z_j)=w_j$ whenever $\displaystyle z_1,z_2,z_3,w_1,w_2,w_3\in\mathbb C-\{0,1,\infty\},$ so I took a generic transformation given by $\displaystyle w(z)=\frac{az+b}{cz+d},$ so I did $\displaystyle i=\frac{a+b}{c+d},$ $\displaystyle -i=\frac{2a+b}{2c+d}$ and $\displaystyle -1=\frac{3a+b}{3c+d},$ 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
    $\displaystyle i=\frac{a+b}{c+d},$ $\displaystyle -i=\frac{2a+b}{2c+d}$ and $\displaystyle -1=\frac{3a+b}{3c+d},$
    All right. Now write the corresponding linear system on the unknowns $\displaystyle a,b,c,d$ , you'll find solutions of the form $\displaystyle a=\alpha d,\;b=\beta d,\;c=\gamma d$ . So the transformation is $\displaystyle w=\frac{\alpha d z+\beta d}{\gamma dz+d}=\frac{\alpha z+\beta }{\gamma z +1}$ .
    Last edited by FernandoRevilla; Dec 4th 2011 at 01:20 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 $\displaystyle \alpha,\beta,\gamma$ you can easily check if the solution is correct, $\displaystyle w(1)=\ldots=i$ etc.
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