How to find the inverse Laplace transform of a specific function?

Question

How can I find the inverse Laplace transform of the following function?

$\mathcal{L}_s^{-1}\left[\frac{\sin \left(a s^2\right)}{s^2}\right](t)$

where a is a positive constant value.

Answer 1

In this case, because $\sin (s)$ is a periodic function, $\mathcal{L}{(f)} = F(s)$ goes to 0 on an infinite amount of points separated by equal intervals. So it follows that a laplace transform $F(s) \neq 0$ cannot be periodic. Therefore, there is no inverse laplace transform of $\bf \frac{\sin (as^2)}{s^2}$.