Solving Quadratic Problem: Find \omega

[dhris]

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 $$\omega$$, 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:

$$|\sigma|>>|\alpha|$$

as well as some vague statement about $$\nu, \eta$$ being small, the two roots of the quadratic are:

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

and

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

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

Thanks,
dhris

 

[Hurkyl]

Well, what is the exact solution for $\omega$; maybe dwelling upon that will indicate how to come up with those approximations.

 

[dhris]

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?

dhris