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Math Help - Möbius transformation

  1. #1
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    Möbius transformation

    Consider the Möbiustransformation w=z^2. Find the image, by this transformation of:

    a) First quadrant.
    b) The region bounded by x=1, y=1 and x+y=1.

    What I have:

    w=z^2=(x+iy)^2=x^2-y^2+2ixy, so let f(z)=f(x+iy)=x^2-y^2+2ixy, but now I don't know what's next. At least for part a) I have x,y\ge0 and for part b) I have a triangle with vertices (0,0), (0,1) and (1,0), but I don't know how to solve the problem though.

    Any help appreciated!!
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  2. #2
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    Re: Möbius transformation

    Can anybody help please?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Re: Möbius transformation

    Hint Write the transformation in the form \begin{Bmatrix} u=x^2-y^2\\v=2xy\end{matrix} and find previously the image of the boundaries.
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  4. #4
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    Re: Möbius transformation

    You mean writing w(x+iy)=u+vi ? For part a) I have w(0)=0, but for part b) I have a triangle, it's just like putting the coordinates on w ? I still don't get how to solve the problem.
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