finding the conformal mapping and plotting complex conformal mapping in Matlab?

Discussion in 'MATLAB' started by Mike, Aug 29, 2006.

  1. Mike

    Mike Guest

    Greetings everybody,

    I am wondering if there is a way to find out the resultant curve for
    the following conformal mapping:

    w= (1-i *z) / (1- i * x * z)

    where z is on the horizontal line a*i -infinity to a*i + infinity, and
    "i" is the imaginary unit, "x" is a parameter.

    I am guessing that this curve must be a circle, but I want to locate
    its center and radius.

    Also it would be very helpful to visualize the conformal mapping in
    Matlab. Any command for that?

    Thanks
     
    Mike, Aug 29, 2006
    #1
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  2. Good guess. A fractional linear transformation such as this one
    maps (straight lines and circles) to (straight lines and circles).
    Since a straight line or circle is determined by three points on it,
    all you have to do now is substitute three values for z. I might
    suggest z=0 and z=infinity as two of the values. You could also
    use the fact that (assuming a and x are real) your curve must be
    symmetric about the real axis, because changing z from t+ia to
    -t+ia takes w to its complex conjugate.

    Robert Israel
    Department of Mathematics http://www.math.ubc.ca/~israel
    University of British Columbia Vancouver, BC, Canada
     
    Robert Israel, Aug 29, 2006
    #2
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  3. Mike

    Mike Guest

    Yes, but how to obtain the center and the radius?
     
    Mike, Aug 29, 2006
    #3
  4. Did you try following my hint? What points did you get?

    Robert Israel
    Department of Mathematics http://www.math.ubc.ca/~israel
    University of British Columbia Vancouver, BC, Canada
     
    Robert Israel, Aug 29, 2006
    #4
  5. Mike

    Mike Guest

    I've found the circle using your hint that the curve must be symmetric
    about the real axis. thanks a lot Robert! You are smart.
     
    Mike, Aug 29, 2006
    #5
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