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Math Help - Transformation

  1. #1
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    Transformation

    The transformation T from the z-plane, where  z = x + iy, to the w-plane, where...

    w=\frac{z+i}{z}, \ z \ne 0

    (a) The transformation T maps the points on the line with equation y=x in the z-plane, other than (0, 0), to points on a line l in the w-plane. Find a cartesian equation of l.

    (b) Show that the image, under T, of the line with equation x+y+1=0 in the z-plane is a circle C in the w-plane, where C has cartesian equation u^2+v^2-u+v=0.

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  2. #2
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    Quote Originally Posted by Air View Post
    The transformation T from the z-plane, where  z = x + iy, to the w-plane, where...

    w=\frac{z+i}{z}, \ z \ne 0

    (a) The transformation T maps the points on the line with equation y=x in the z-plane, other than (0, 0), to points on a line l in the w-plane. Find a cartesian equation of l.
    The complex number z has an argument of \frac{ \pi}{4} for \Im (z) > 0 and - \frac{ 3 \pi }{4} for \Im (z) < 0.

    Consider w - 1 which is \frac{i}{z} therefore \arg(w -1) = \arg \left( \frac{i}{z} \right)

    \Rightarrow \arg(w -1) = \arg(i) - \arg(z)
    \Rightarrow \arg(w -1) = \frac{\pi}{2} - \arg(z)

    for \Im (z) > 0 \arg(w -1) = \frac{\pi}{2} - \frac{\pi}{4} \ \ \Rightarrow \ \ \arg(w -1) = \frac{\pi}{4}

    for \Im (z) < 0 \arg(w -1) = \frac{\pi}{2} + \frac{3 \pi}{4} \ \ \Rightarrow \ \ \arg(w -1) = \frac{5 \pi}{4}

    You should be able to pull out an equation form that.

    Bobak
    Last edited by bobak; June 10th 2008 at 02:32 PM.
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