1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.

    Thanks!
     
  2. jcsd
  3. Jul 28, 2012 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    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

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    No worries :-)
    Sometimes the trick is starting without knowing whe re you are going.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Mapping unit circle from one complex plane to another
  1. Unit Circle (Replies: 3)

  2. Complex mapping (Replies: 4)

Loading...