I am trying to solve a system of partial differential equations arising from a mathematical physical problem using the Laplace transform approach. In order to retrieve the solution in the time domain, one has to perform an inverse Laplace transformation.
The function of interest is given by $$ F(p) = \frac{\lambda}{p} \, \tanh (\lambda) \, , \text{ where } \lambda=\left(\frac{p(1+p)}{2(1+2p)}\right)^{1/2} \, . $$
Clearly, $F(p)$ is well behaved around $p=0$, since $\lim\limits_{p\to 0} F(p) = \frac{1}{2}$. However, $F(p)$ at $p=-\frac{1}{2}$ is undefined. In order to apply the inverse transformation, one has usually to perform a complex integration by choosing an abscissa of convergence $\sigma>0$ for which the contour is located to the right of all singularities.
I was wondering whether someone here, who has some previous experience or knowledge with such integral transforms try to be of help and tell whether an analytical (or numerical) treatment is possible.
Your hints and suggestions are most welcome.
Thank you,