Running through a complex math derivation of plasma frequency

[Suzanne Rosenzweig]

TL;DR

Running through a derivation of plasma frequency, not sure why we can drop imaginary part.

Background of problem comes from Drude model of a metal (not necessary to answer my problem but for the curious): Consider a uniform, time-dependent electric field acting on a metal. It can be shown that the conductivity is $$\sigma = \frac{\sigma_0}{1-i\omega t}$$ where $$\sigma_0 = \frac{ne\tau^2}{m}.$$ From Maxwell's equations, we can get a wave equation of the form $$-\nabla^2\textbf{E} = \frac{\omega^2}{c^2}\epsilon(\omega)\textbf{E}$$ where the dielectric constant is $$\epsilon(\omega) = 1+\frac{4\pi i\sigma}{\omega}.$$ We can substitute into this equation the first two definitions provided ($\sigma$ and $\sigma_0$) and take the limit that $\omega\tau \gg 1$ to find, to a first approximation, that $$\epsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2}$$ where $$\omega_p^2 = \frac{4\pi ne^2}{m}.$$ Where I need help: The problem I have is that when I run through my derivation (plugging the expression for $\sigma$ and $\sigma_0$ into the equation for $\epsilon(\omega)$ and taking it to a first approximation since $\omega\tau\gg1$), I can get the real part to agree with this expression but I can't reason why we can drop the imaginary part. The imaginary part I wind up with looks like this: $$\frac{4\pi i ne^2\tau}{m\omega(1+\omega^2\tau^2)}.$$ Why can we drop this without consequence?

Note: I put this in the Math forum because it's purely the mathematical reasoning I need help with but I provided the physics background for those interested.

 

[fresh_42]

I get a different formula, since I did not set $\tau = t$ or switched the square in $\sigma_0$ to $e$. So it is guesswork now. But with the original formulas, I get an imaginary part
$$
\omega \varepsilon = \text{ real part } + i \cdot \left( 4\pi \dfrac{n}{m} e \dfrac{\tau^2}{1+t^2\omega^2} \right)
$$
which tends to $0$ if $t\omega \gg 1$.