Inverse Laplace transform of $f(s)=\frac{1}{s\sinh^2(c\sqrt{s})}$

Question

I want to calculate inverse Laplace transform of $f(s)=\frac{1}{s\sinh^2(c\sqrt{s})}$, where $s$ the is Laplacian variable.

The function has one pole at $s=0$, for which the residue can be easily found, and infinitely many poles at $c\sqrt{s} = n\pi i$. I have some problems finding the residues of those poles because seem to be of order of $2$? How can I solve the second part of the problem?

Thanks in advance.

Answer 1

$$f(s)=\frac{1}{s \sinh ^2\left(c \sqrt{s}\right)}$$ Integrate f(s) as a function a: $$\text{Int}(f(a,s))=\int \frac{1}{s \sinh ^2\left(a c \sqrt{s}\right)} \, da=-\frac{\coth \left(a c \sqrt{s}\right)}{c s^{3/2}}+\text{c1} $$ Using this identity and substituting:

$$\coth (s)=\frac{1}{s}+\sum _{k=1}^{\infty } \frac{2 s}{\pi ^2 \left(\frac{s^2}{\pi ^2}+k^2\right)}$$

$$\text{Int}(f(a,s))=-\frac{1}{a c^2 s^2}-\sum _{k=1}^{\infty } \frac{2 a}{\pi ^2 s \left(k^2+\frac{a^2 c^2 s}{\pi ^2}\right)}+\text{c1}$$ Using Inverse Laplace Transfrom: $$\mathcal{L}_s^{-1}[\text{Int}(f(a,s))](t)=\mathcal{L}_s^{-1}\left[-\frac{1}{a c^2 s^2}-\sum _{k=1}^{\infty } \frac{2 a}{\pi ^2 s \left(k^2+\frac{a^2 c^2 s}{\pi ^2}\right)}+\text{c1}\right](t)$$

$$\text{Int}(F(a,t))=-\frac{t}{a c^2}-\sum _{k=1}^{\infty } \left(\frac{2 a}{k^2 \pi ^2}-\frac{2 a e^{-\frac{k^2 \pi ^2 t}{a^2 c^2}}}{k^2 \pi ^2}\right)+\text{c1} \delta (t)$$

We differentiate for a and a=1: $$\frac{\partial \text{Int}(F(a,t))}{\partial a}=\frac{\partial }{\partial a}\left(-\frac{t}{a c^2}-\sum _{k=1}^{\infty } \left(\frac{2 a}{k^2 \pi ^2}-\frac{2 a e^{-\frac{k^2 \pi ^2 t}{a^2 c^2}}}{k^2 \pi ^2}\right)+\text{c1} \delta (t)\right)$$

$$F(t)=\frac{t}{c^2}-\sum _{k=1}^{\infty } \frac{2}{k^2 \pi ^2}+\sum _{k=1}^{\infty } \frac{2 e^{-\frac{k^2 \pi ^2 t}{c^2}}}{k^2 \pi ^2}+\sum _{k=1}^{\infty } \frac{4 e^{-\frac{k^2 \pi ^2 t}{c^2}} t}{c^2}$$

$$F(t)=-\frac{1}{3}-\frac{t}{c^2}+\frac{2 t \vartheta _3\left(0,e^{-\frac{\pi ^2 t}{c^2}}\right)}{c^2}+\sum _{k=1}^{\infty } \frac{2 e^{-\frac{k^2 \pi ^2 t}{c^2}}}{k^2 \pi ^2}$$

where $\vartheta _3\left(0,e^{-\frac{\pi ^2 t}{c^2}}\right)$: is Jacobi Theta function