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Quadratic Problem

  1. Dec 5, 2003 #1
    Hi, I'm hoping someone out there is going to see something in this problem that I don't because I really don't get it:

    Consider the equation:

    \sigma=(\omega + i \nu k^2)+\frac{\alpha^2}{\omega + i \eta k^2}

    It doesn't really matter what the variables mean, (i^2=-1 of course) but what I really need is to figure out [tex] \omega [/tex], which is complex, as a function of the rest (under a certain approximation). The book I found this in claims that under the following conditions:


    as well as some vague statement about [tex] \nu, \eta [/tex] being small, the two roots of the quadratic are:

    \omega \approx -i \nu k^2 + \sigma + \frac{\alpha^2}{\sigma + i(\eta-\nu)k^2}


    \omega \approx -i \eta k^2 - \frac{\alpha^2}{\sigma}

    I don't know how they came up with this, but it would be really great to find out. Anybody have any ideas?

    Last edited: Dec 5, 2003
  2. jcsd
  3. Dec 5, 2003 #2


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    Well, what is the exact solution for [itex]\omega[/itex]; maybe dwelling upon that will indicate how to come up with those approximations.
  4. Dec 7, 2003 #3
    Thanks, that's what I was doing. I couldn't see how they applied the approximation though, but figured it out soon after I posted. Why does it always happen that way?

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