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Mapping unit circle from one complex plane to another

  1. Jul 28, 2012 #1
    I want to show that if the complex variables ζ and z and related via the relation

    z = (2/ζ) + ζ

    then the unit circle mod(ζ) = 1 in the ζ plane maps to an ellipse in the z-plane.

    Then if I write z as x + iy, what is the equation for this ellipse in terms of x and y?

    Any help would be much appreciated.

  2. jcsd
  3. Jul 28, 2012 #2

    Simon Bridge

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    Welcome to PF;
    ... you mean: "how should I go about finding the equation of the ellipse?" Nobody is going to spoon-feed you the actual answer here - but we can help you find it for yourself.

    You can help us do that by attempting the problem.

    Start out by writing out the relations you know:

    1. z = (2/ζ) + ζ
    2. |ζ| = 1

    3. ζ = γ + iλ
    4. z = x + iy

    5. ... any other relations that must hold true?

    Presumably you can expand 1 and 2 in terms of 3?
    Presumably you can look up the general equation of an ellipse?

    Now where do you get stuck?
  4. Jul 29, 2012 #3
    I think I've got it now..

    Let ζ = u+iv so u²+v²=1 because |ζ| = 1

    2/ζ + ζ = 2 / (u+iv) + (u+iv) = 2(u−iv) / (u²+v²) + (u+iv) = 3u−iv

    ∴ x+iy = 3u−iv and so u=x/3, v=−y

    From u²+v² = 1 this yields (x/3)²+y² = 1, an ellipse

    Thanks for your help
  5. Jul 29, 2012 #4

    Simon Bridge

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    No worries :-)
    Sometimes the trick is starting without knowing whe re you are going.
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