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

  1. Sep 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Could anyone help me please?
    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
     
  2. jcsd
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