TITLE: Laplace transformation-exponential
QUESTION [0 upvotes]: 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.

REPLY [0 votes]: $$\mathcal{L}_{\alpha }^{-1}\left[\exp \left(-\alpha  \sqrt{p^2+\beta ^2}\right)\right](t)=\delta (t-\alpha )-\frac{\alpha  \beta  \ H (t-\alpha ) J_1\left(\sqrt{t^2-\alpha ^2} \beta \right)}{\sqrt{t^2-\alpha ^2}}$$
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.
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