)\equiv \frac{1}{\pi }\frac{\Delta}{(\omega
-\bar \omega )^2+\Delta ^2}=\frac{1}{2\pi i}\left\{
\frac{1}{\omega -(\bar \omega +i\Delta )}-\frac{1}{\omega -(\bar
\omega -i\Delta )}\right\} \ .$$ The right-most expression for $g_L(\omega )$ makes clear that, when the previously real variable $\omega $ is analytically continued into the complex plane, the function $g_L(\omega )$ results from the sum of two pole contributions, one at $\omega
=\bar \omega +i\Delta $, and one at $\omega =\bar \omega -i\Delta
$. The quantity $\bar \omega $ represents the average frequency of the distribution, while $\Delta $ represents the width of the distribution. Consideration of the Lorentzian will be particularly useful to us because the integral (32) can be explicitly evaluated, and also because our numerical experiments in Sec. IV will be for the case of a Lorentzian frequency distribution function.

As a first illustration we consider the $n=-1$ term which results in an echo at $t=2\tau $. We first evaluate $D(s)$ by inserting the pole-form for $g_L(\omega )$ into Eq. (18) and closing the integration path with a large semicircle of radius approaching infinity. This yields a single residue contribution to $D(s)$, $$D(s)=1-\frac{K}{2}[s+\xi +i(\bar \omega -i\Delta )]^{-1} \ .$$ Note that the solution of $D(s)=0$ occurs at $$s=-i\omega -\left(\xi +\Delta -\frac{K}{2}\right) \ 