$$f^{(2)}_{1,e}(\omega ,\tau ^+)=d_0d_1h_2h_1^*\frac{\exp (i\omega
\tau -\xi \tau )}{D^*[-(i\omega +\xi )]}-d_0d_1\sum _{|n|\geq
2}nh_nh^*_{n-1}\exp [-(in\omega +n^2\xi )\tau ] \ ,$$ where we have used $h_0=0$,  $h_n=h^*_n$, $f^{(1)}_{-1}=f_1^{(1)*}$, and the first term on the right side of Eq. (33) corresponds to $n=-1$ in Eq. (25). Putting Eq. (33) into Eq. (32), we see that we have an integral of a sum over terms with exponential time variations of the form $$\exp\{-i\omega [t-(1-n)\tau ]\}\exp \{ -\xi [t+(n^2-1)\tau \} \ .$$ Considering the first exponential in Eq. (34), we see that, for large values of $|t-(1-n)\tau |$, there is rapid oscillation of the integrand with $\omega $, and the integral can therefore be expected to be near zero. However, such rapid oscillation is absent near the times $t=(1-n)\tau $, at which a large value of $R_e^{(2)*}$ will occur. Since $t>\tau $, the relevant times occur for $n<-1$; e.g., for $n=-1$, we get an echo at $t=2\tau$; for $n=-2$, we get an echo at $t=3\tau $; etc. Therefore, we henceforth replace the summation over $|n|\geq 2$ in Eq. (33) by a summation from $n=-\infty$ to $n=-2$.

E. Evaluation for Lorentzian frequency distribution functions
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We now consider the case of a Lorentzian frequency distribution, $$g(\omega )=g_L(\omega 