# Proof of the Laplace transformation of the Bessel function with square argument

1. Sep 28, 2012

### Domdamo

1. The problem statement, all variables and given/known data

I would like to know the proof of the following Laplace transform pair:

2. Relevant equations

\mathcal{L}_{t \rightarrow s} \left\{ J_0 \left( a\sqrt{t^2-b^2} \right) \right\}=\frac{e^{-b\sqrt{s^2+a^2}}}{\sqrt{s^2+a^2}}

3. The attempt at a solution

I tried to prove with the series method or the method of the differential equation according to (ref. 2) but I failed.

This identity can be found in these books:
(1) Bateman H, Erdélyi A; Tables of Integral Transforms, Vol. I.; 1954; pp. 191., eq. (9)
or
(2) Spiegel M R; Schaum's outline of Laplace Transform; 1965; pp. 249., eq. (71) and pp. 23, Problem 34 Bessel functions