hi,

I have a function $X(t)$ whose Laplace transform $\hat{X}(s)$ has a unique pole of largest real part $x_0$, which is a real number. I am able to show that for each $t$, $X(t)$ is a convergent sum of the residues of $\hat{X}(s)e^{st}$, over its poles. If $x_0$ is a pole of order $m$, is it true in general that $X(t)/(t^{m-1}e^{x_0t})$ converges as $t\rightarrow\infty$?