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I have proved (through educated guess-work and checking analytically) the following identity

[itex]\int\limits_0^\infty\int\limits_0^\infty s_1 \exp\left(-\gamma s_1\right) s_2 \exp\left(-\gamma s_2\right) J_0\left(s_1r_1\right) J_0\left(s_2r_2\right) ds_1ds_2 = \frac{\gamma}{\left(r_1^2+r_2^2+\gamma^2\right)^{5/2}} [/itex]

[itex]J_0[/itex] is a Bessel function of the first kind.

However I can't find this identity in any book and therefore stuck in how to give a rigorous reference (rather than say I guessed it, and I was right).

However I have found a related single integral identity (that IS in a book) and wonder if it could be used to prove the above.

[itex]\int_0^\infty \! s \exp\left(-\gamma s\right)J_0\left(sr\right)ds = \frac{\gamma}{\left(r^2+\gamma^2\right)^{3/2}}[/itex]

I have tried Mathematica, Maple, Matlab,... you name it but to no avail. I've trawled all the usual book suspects (of integral tables) but also.... zilch.

Any ideas of where to start or any leads here?

thanks in advance.

Paul