$\mathcal L^{-1}\left(\frac 1 s \exp\left(-\sqrt{\frac{as}{s+b}}\right)\right)$ using contour integration. Where a and b are constants.

Question

How to evaluate $\mathcal L^{-1}\left(\frac 1 s \exp\left(-\sqrt{\frac{as}{s+b}}\right)\right)$ by using contour integration or some other methods. Where a and b are constants

Answer 1

Note that:

$$\exp (x)=\sum _{j=0}^{\infty } \frac{x^j}{j!}$$ so, $$\color{red}{\mathcal{L}_s^{-1}\left[\frac{\exp \left(-\sqrt{\frac{a s}{s+b}}\right)}{s}\right](t)}=\mathcal{L}_s^{-1}\left[\sum _{j=0}^{\infty } \frac{\left(-\sqrt{\frac{a s}{s+b}}\right)^j}{s j!}\right](t)=\sum _{j=0}^{\infty } \mathcal{L}_s^{-1}\left[\frac{(-1)^j a^{j/2} s^{-1+\frac{j}{2}} (b+s)^{-\frac{j}{2}}}{j!}\right](t)=\color{red}{\sum _{j=0}^{\infty } \frac{(-1)^j a^{j/2} L_{-\frac{j}{2}}(-b t)}{j!}}$$

where: $L_{-\frac{j}{2}}(-b t)$ is the Laguerre polynomial.

Does this sum have a closed form ? Possible No.