Find $L^{-1}{\frac{{{e^{-\sqrt{p}}}}}{p}}$

Question

Find $$L^{-1}{\frac{{{e^{-\sqrt{p}}}}}{p}}$$. where $L^{-1}$ denotes the laplace inverse .

I tried expanding the exponential but got stuck in the middle.

Answer 1

All done in Mathematica:

$$\mathcal{L}_p^{-1}\left[\frac{\exp \left(-\sqrt{p}\right)}{p}\right](t)=\\\mathcal{L}_p^{-1}\left[\frac{\sum _{j=0}^{\infty } \frac{\left(-\sqrt{p}\right)^j}{j!}}{p}\right](t)=\\\sum _{j=0}^{\infty } \mathcal{L}_p^{-1}\left[\frac{(-1)^j p^{-1+\frac{j}{2}}}{j!}\right](t)=\\\sum _{j=0}^{\infty } -\frac{2 (-1)^j t^{-\frac{j}{2}}}{j j! \Gamma \left(-\frac{j}{2}\right)}=\sum _{j=0}^{\infty } \frac{(-1)^j t^{-\frac{j}{2}}}{\Gamma \left(1-\frac{j}{2}\right) \Gamma (1+j)}=\\1-\text{erf}\left(\frac{1}{2 \sqrt{t}}\right)=\text{erfc}\left(\frac{1}{2 \sqrt{t}}\right)$$