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Writing a function of x and y as a function of z

  1. Mar 9, 2013 #1
    Is there a general method for this?
    I remember in complex analysis going the other direction; that is, a function of z becoming a function of x and y.
    But I need to turn a complex function in a function of just z, for the purpose of integrating with respect to z.
    In particular, the function is
    x^2/2-cx-y^2/2+ i(x-c)y where c is a constant.
  2. jcsd
  3. Mar 9, 2013 #2


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    When you say z, are you referring to z, as in the third coordinate of an x,y,z system or the complex variable z, as in z= x + iy?
  4. Mar 9, 2013 #3
    complex variable z, as in z=x+iy.
  5. Mar 9, 2013 #4
    In fact I now see how to write this in terms of z but it is from recognizing the form. Is there a more general method?
    Please no one reply with "well if you recognize the form in this particular equation.", because again, I already see it.
    General method please?
  6. Mar 9, 2013 #5


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    Hi, try:

    x= (z+ z^)/2 , where z^ is the conjugate, i.e. if z=x+iy, then z^= x-iy

    y=( z- z^)/2i

    It comes from z=x+iy
  7. Mar 9, 2013 #6
  8. Mar 9, 2013 #7


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    Be aware that you cannot necessarily write a function f(x,y) solely as a function of ##z = x + i y##. You may find that the complex conjugate ##z^\ast = x - iy## does not cancel out of your expression.

    You will only be left with a function of z (and not z*) if your function is analytic in the complex plane.
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