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Simultaneous nonlinear equations

  1. Sep 7, 2008 #1
    Hi. I have the following two equations

    [tex]S_{21}=\frac{(1-\Gamma^2)z}{1-z^2\Gamma^2}[/tex]
    [tex]S_{11}=\frac{(1-z^2)\Gamma}{1-z^2\Gamma^2}[/tex]

    How would you go about solving these equations? I want to avoid square roots because they make the results ambiguous.

    I myself have found that

    [tex]z=\pm\sqrt{\frac{\Gamma-S_{11}}{\Gamma-S_{11}\Gamma^2}}[/tex]

    but a better solution is

    [tex]z=\frac{S_{21}}{1-S_{11}\Gamma}[/tex]

    because it avoids the sign ambiguity. Yet another good solution is

    [tex]z=\frac{(S_{11}+S_{21})-\Gamma}{1-(S_{11}+S_{21})\Gamma}[/tex]

    but I have no clue how to arrive at these results. Any suggestions?
     
  2. jcsd
  3. Sep 8, 2008 #2
    Are there any general methods for solving nonlinear equations analytically? I only know of the substitution method and then applying the quadratic forumla.
     
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