# Analytic verification of Kramers-Kronig Relations

Septim

## Homework Statement

Show that the real and imaginary parts of the following susceptibility function satisfy the K-K relationships. Use the residue theorem.
$$\chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega}$$

## Homework Equations

The Kramers-Kronig relations are

$$\chi_r(\omega) = \frac{1}{\pi} P \int_{-\infty}^{\infty} d\bar{\omega} \frac{\chi_i(\bar{\omega})}{\bar{\omega}-\omega}$$
$$\chi_i(\omega) = -\frac{1}{\pi} P \int_{-\infty}^{\infty} d\bar{\omega} \frac{\chi_r(\bar{\omega})}{\bar{\omega}-\omega}$$

## The Attempt at a Solution

The problem is that my complex calculus is pretty rusty and I do not know which poles contribute exactly. There are 5 poles in total 4 from the susceptibility function and 1 from the denominator(see the expressions please). The poles are
$$\pm \gamma \pm i\frac{\sqrt{4\omega_0^2-\gamma}}{2}$$

I calculated the residues that are on the negative imaginary $\omega$ plane on Mathematica and they turned out to be zero except the pole at $\omega$ which is the value of the integral using the principal rule in Mathematica. Is this analytically tractable easily? I appreciate any assistance you provide.

Many thanks,

$$\chi(\omega) = \frac{\omega_{p}^2}{(\omega_0^2-\omega^2)+i\gamma\omega}$$