I'm trying to get the inverse Laplace transform of the following transfer function:
$$ \mathcal{L}_s^{-1}\bigl[F(s)\bigl] =\mathcal{L}_s^{-1}\biggl[\frac{\tanh\sqrt{s}}{\sqrt{s}-\tanh \sqrt{s}}\biggl] = \mathcal{L}_s^{-1}\biggl[\frac{1}{\sqrt{s}\coth \sqrt{s} -1}\biggl] $$
I numerically verified that the denominator has an infinite zeros, $s_j \in \mathbb{R}_{\leq0}$, so I expect a solution of the type:
$$ \mathcal{L}_s^{-1}\bigl[F(s)\bigl]= \sum_{j}^\infty \operatorname{Res} \left(F(s)\exp [st],s_j \right) $$
However, I am struggling with the evaluation of the residues, starting from $s_0 = 0$ : $$ \operatorname{Res} \left(F(s)\exp [st],s_0 \right) = \lim_{s~\to~0} \biggl[\frac{se^{st}}{\sqrt{s}\coth \sqrt{s} -1}\biggl] $$
What is the best method to evaluate this and other $s_j$ limits?
