  Given $\beta>0$ and $\delta>0$, the function $t^{-\beta}$ may be approximated
for $t$ in a compact interval $[\delta,T]$ by a sum of terms of the form
$we^{-at}$, with parameters $w>0$ and $a>0$. One such an approximation, studied
by Beylkin and Monz\'on, is obtained by applying the trapezoidal rule to an
integral representation of $t^{-\beta}$, after which Prony's method is applied
to reduce the number of terms in the sum with essentially no loss of accuracy.
We review this method, and then describe a similar approach based on an
alternative integral representation. The main difference is that the new
approach achieves much better results before the application of Prony's method;
after applying Prony's method the performance of both is much the same.
