can anyone tell me how can I seek the inverse Laplace transform of equation $F(s)=\frac{p}{s}-\frac{p}{s}\frac{sinh(a\sqrt{s})}{sinh(b\sqrt{s})}$, where $p$, $a$, and $b$ are positive constants independent on $s$.
I found in a hand book that $L^{-1}\{\frac{1}{s-iw}\frac{sinh(a\sqrt{s})}{sinh(b\sqrt{s})}\} = \frac{sinh(a\sqrt{wi})}{sinh(b\sqrt{wi})}e^{iwt} + 2\pi \sum_{n=1}^\infty \frac{n (-1)^n sin(n\pi a/b)}{n^2 \pi^2 + iwb^2}e^{-n^2 \pi^2 t/b}$, but I don't know how to handle the first term since $w=0$.
Many thanks to your help. A detailed explanation will be highly appreciated, since I'm new in Laplace transform and I really don't have much knowledge in complex analysis.
