Inverse Laplace transform of $\exp(-1/\sqrt{s})$

Question

I'm looking for the inverse Laplace transform of:

$$F(s) = \exp(-1/\sqrt{s}).$$

Does the inverse Laplace transform exist? Do you have a reference in which this transform is given?

Answer 1

Since for every $\alpha>0$: $$\mathcal{L}^{-1}\left(s^{-\alpha}\right)=\frac{x^{\alpha-1}}{\Gamma(\alpha)}\tag{1}$$ and: $$\exp\left(-s^{-1/2}\right) = \sum_{k=0}^{+\infty}\frac{(-1)^k}{k!}\,s^{-k/2} \tag{2}$$ we have: $$\mathcal{L}^{-1}\left(\exp\left(-s^{-1/2}\right)\right) = \delta(x)+\sum_{k=1}^{+\infty}\frac{(-1)^k}{k!}\cdot\frac{x^{k/2-1}}{\Gamma(k/2)}.\tag{3}$$