Laplace transformation-exponential

Question

I need to get the Inverse Laplace Transform in the form of following function:

$$e^{-\alpha \sqrt{p^2+\beta^2}}$$

α and β being some parameters which α is positive.

Answer 1

$$\mathcal{L}_{\alpha }^{-1}\left[\exp \left(-\alpha \sqrt{p^2+\beta ^2}\right)\right](t)=\delta (t-\alpha )-\frac{\alpha \beta \ H (t-\alpha ) J_1\left(\sqrt{t^2-\alpha ^2} \beta \right)}{\sqrt{t^2-\alpha ^2}}$$

where $J_1(t)$ is Bessel function of the first kind.

where $\ H (t-\alpha )$ is Heaviside theta function.

where $\delta (t-\alpha )$ is Dirac delta function.

From Book. Table A.5.1 example 191.