I have this integral related to a Laplace transform and I was wondering if anyone knows of a clever way to derive it. I know we usually look these up in a table, but this form is not in a table I have. The transform is:
$\mathcal{L}\left\{ \frac{\sin 1/t}{\sqrt{t}} \right\}(s)=\int\limits_{0}^{\infty}\frac{\sin(1/t)}{\sqrt{t}}e^{-st}\,\mathrm dt, \;\ s>0.$
I ran it through Maple and it gave me $\sqrt{\frac{\pi}{s}}\sin(\sqrt{2s})e^{-\sqrt{2s}}.$
A rather simplistic looking solution, but how to derive?
I did notice that the $\frac{1}{t^{\frac{1}{2}}}=\sqrt{\frac{\pi}{s}},$
since the Laplace of $t^{k}=\frac{\Gamma(k+1)}{s^{k+1}}, \;\ k>-1.$ This would mean
$t^{\frac{-1}{2}}=\frac{\Gamma(\tfrac12)}{\sqrt{s}}=\sqrt{\frac{\pi}{s}}.$
Perhaps it is too difficult to do by hand. I just thought it was interesting how to derive this if possible.
Thanks very much for your time, interest, and expertise.