I need to evaluate (if it exists) the inverse Laplace transform of the following complex function $F(s)$:
$$
F(s)=\sqrt{\frac{s}{a}} J_{1}(\sqrt{as})
$$
where $J_{1}(\cdot)$ is the Bessel function of first kind and first order.
Does anyone have any suggestion?
I know that an inverse Laplace transform exists for a similar expression, i.e.,
$$
L \left\{ \frac{1}{4t^2 } e^{-\frac{a}{4t}} \right\} = \sqrt{\frac{s}{a}} K_{1}(\sqrt{as})
$$
where $K_{1}(\cdot)$ is the modified Bessel function of second kind and first order.
Thanks!
Best Answer
I guess you're struggling with Voronoi's improvement of the Gauss circle problem. If you're interested, I am writing a section about it in my notes, it is not finished yet but it will probably be before the new year (2019). Anyway, you may consider Bessel's differential equation defining $J_1$, or directly the Maclaurin series of $J_1$, and discover that
$$\color{red}{\mathcal{L}}\left(\sum_{n\geq 0}\frac{(-1)^n a^{n} (s)^{n+1}}{ 2^{2n+1} n!(n+1)!}\right) = \frac{1}{2x^2} e^{-\frac{a}{4x}}$$ but the series is associated to an entire function, so its inverse Laplace transform is simply not defined, unless you're fine with a distributional identity $\mathcal{L}^{-1}(s^n)(x)=\delta^{(n)}(x)$.