Estimating the damping coefficient of a wave assuming a very small ratio

[loto]

Homework Statement

The original problem is determining the dispersion relation of an ordinary wave in plasma damped by collisions. That part was easy enough but the next part is to find the damping rate (-Im(ω)) of the wave, assuming k is real and $\nu << \omega$ where $\nu$ is the collision frequency.

Homework Equations

I've found the dispersion relation to be:

$\omega^2 - k^2c^2 = \frac{\omega_{pe}^2(1-i\frac{\nu}{\omega})}{(1 + \frac{\nu^2}{\omega^2})}$

And we are told the damping rate is:
$\gamma = \frac{\omega_{pe}^2\nu}{\omega^2 2}$

The Attempt at a Solution

Since we want only the negative of the imaginary part:

$\gamma = \frac{\nu\omega_{pe}^2}{\omega^2 (1+\frac{\nu^2}{\omega^2 })}$

However, I can't think of an approximation that would give me that factor of 1/2. Series expansion or small number approximations don't seem to do it. If anyone has ideas, I'd just like a push in the right direction?

 

[AlephZero]

Apologies if this is irrelevant, but some people write the equation for damped oscillations as
$\ddot x + b\omega \dot x + \omega^2 x = 0$
and other people write it as
$\ddot x + 2 b\omega \dot x + \omega^2 x = 0$
I deliberately wrote $b$ in those equations rather than the usual greek letters, because I don't know what notation convention is used in plasma dynamics!

Assuming you got the first part of the question right, is that where your factor of 2 has come from?