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Complex Analysis: Nonlinear system

  1. Sep 16, 2004 #1
    Here's a problem I ran into in complex analysis. Given z = x + iy and w = u + iv, I need to find all w such that w² = z. It reduces to solving this system:

    x = u² - v²
    y = 2uv

    My professor mentioned that we should try to deal with the problem in at least two cases: y = 0, and y does not equal 0 (perhaps the second case should also be split up into a few cases). But beyond the y = 0 case, I'm stuck. Can anyone help?
     
  2. jcsd
  3. Sep 16, 2004 #2

    Tide

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    Why don't you try writing the complex numbers in complex exponential form form?
     
  4. Sep 17, 2004 #3

    arildno

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    The second equation implies:
    [tex]y^{2}=4u^{2}v^{2}[/tex]
    which is equivalent to (when substituting from the first equation):
    [tex]4(v^{2})^{2}+4x(v^{2})-y^{2}=0[/tex]
    Or:
    [tex]v^{2}=\frac{-4x\pm\sqrt{16x^{2}+16y^{2}}}{8}=\frac{-x\pm{r}}{2},r=\sqrt{x^{2}+y^{2}}[/tex]
    Comments:
    1. Clearly, only non-negative values are acceptable for [tex]v^{2}[/tex]
    This means that we, for all x, have: [tex]v^{2}=\frac{r-x}{2}[/tex]
    2. Even more important, the equations:
    [tex]y=2uv[/tex],[tex]y^{2}=4u^{2}v^{2}[/tex]
    are not equivalent; hence your answers may contain false solutions; ie, you must substitute what you get into your original system in order to determine the actual solutions.
     
    Last edited: Sep 17, 2004
  5. Sep 18, 2004 #4
    Thanks arildno! I'll try your method; it looks familiar because the prof also mentioned something about a ± popping up in the problem.
     
  6. Sep 18, 2004 #5

    arildno

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    Good luck, arunma!
    However, if you've learnt about complex exponentials, it is a quite instructive additional exercise to discover the equivalence of Tide's and my own approach..
     
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