Calculate inverse Laplace transform of $\exp(\sqrt{s^2-r^2})$

Question

I have some trouble with the calculation of the inverse Laplace transform $e^{-k\sqrt{s^2-r^2}}$ , $k\geq0$, $r$ is known. ##

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## And I believe it has some relation with the inverse Laplace transform of $\frac{e^{-k\sqrt{s^2-r^2}}}{\sqrt{s^2-r^2}}$ , but I find it hard to deal with the modified Bessel function $I_0$.

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Hope some one can help me :).

Answer 1

In the book I mentioned in the comments there there are identities for $s=\sqrt{p^2-\alpha^2}$ where $p$ is the variable being inverse transformed. You should be able to relate the expression you know to the one you don't know through this.