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Math Help - Complex function : composition of a translation, rotation, etc

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    MHF Contributor arbolis's Avatar
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    Complex function : composition of a translation, rotation, etc

    I don't know how to approach the following problem : Write the transformation T(z)=\frac{i-z}{i+z} as a composition of translations, rotations, dilatations and inversions. Use the result to find the image of the semi-plane y>0.

    My attempt : My idea was to write T(z) in the form re^{i \pi \theta} but I didn't reach anything important I believe.
    So I guess there's a better way to approach the problem. I'd like to know it/them.
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    Quote Originally Posted by arbolis View Post
    I don't know how to approach the following problem : Write the transformation T(z)=\frac{i-z}{i+z} as a composition of translations, rotations, dilatations and inversions. Use the result to find the image of the semi-plane y>0.

    My attempt : My idea was to write T(z) in the form re^{i \pi \theta} but I didn't reach anything important I believe.
    So I guess there's a better way to approach the problem. I'd like to know it/them.
    If you are interested in conformal mapping you can read the entire detailed thread here.

    \frac{i-z}{i+z} = -\frac{z-i}{z+i} = - \frac{z+i-2i}{z+i} = -1 + \frac{2i}{z+i}

    Let f: z\mapsto z+i, let g:z\mapsto \tfrac{1}{z}, let h: z \mapsto i, let F: z\mapsto 2z, let G:z\mapsto z-1. Thus, f is translation, g is inversion, h is rotation (by \tfrac{\pi}{2}), F is dilation, and G is translation.

    However, G\circ H \circ h\circ g \circ f: z\mapsto \tfrac{i-z}{i+z}.
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    MHF Contributor arbolis's Avatar
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    Thanks a lot TPH! (I'll take some time to read through the link... seems really interesting. I don't know if my course will introduce me conformal mapping, it is made for physicists.)
    Just a little question... why did you bother to rewrite \frac{i-z}{i+z} in the form -1 + \frac{2i}{z+i}?
    I see that you got rid of a z over z form... is it a general way to solve problem like these?


    Quote Originally Posted by ThePerfectHacker View Post
    If you are interested in conformal mapping you can read the entire detailed thread here.

    \frac{i-z}{i+z} = -\frac{z-i}{z+i} = - \frac{z+i-2i}{z+i} = -1 + \frac{2i}{z+i}

    Let f: z\mapsto z+i, let g:z\mapsto \tfrac{1}{z}, let h: z \mapsto i, let F: z\mapsto 2z, let G:z\mapsto z-1. Thus, f is translation, g is inversion, h is rotation (by \tfrac{\pi}{2}), F is dilation, and G is translation.

    However, G\circ H \circ h\circ g \circ f: z\mapsto \tfrac{i-z}{i+z}.
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    Quote Originally Posted by arbolis View Post
    I see that you got rid of a z over z form... is it a general way to solve problem like these?
    Yes, we have to clear the z in the numerator. Because the basic inversion is 1/z, it has no z in numerator. Thus, we have to bring it into that form.
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    Yes, we have to clear the z in the numerator. Because the basic inversion is 1/z, it has no z in numerator. Thus, we have to bring it into that form.
    Thanks. I'm infinitely thankful to you.
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    Quote Originally Posted by arbolis View Post
    Thanks. I'm infinitely thankful to you.
    You can bow before my feet and say, "Oh Master how much I love thee".
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    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    You can bow before my feet and say, "Oh Master how much I love thee".

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