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I'm try to derive a closed form of the following double integral: $\int\limits_0^x {\int\limits_0^x {{e^{ - {K_1}uv}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)du} dv}$; where $K_1$ is a constant. Do you have any suggestions? Thank you very much.

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Do you have some background for this question. Also (maybe related), why do you tag it with integral-transform. –  Fabian Jan 10 '13 at 13:36
    
My experience with integrals like this is that they have no "closed" form unless $x = \infty$, and that's for the single integral. –  Ron Gordon Jan 10 '13 at 13:38
    
@rlgordonma: 2D and Bessel functions sometimes help each other... –  Fabian Jan 10 '13 at 13:39
    
@Fabian: I agree. This case, however, is equivalent to an integral of a Bessel-Gaussian form. With most special functions like this, unless the integration bounds are branch points or $\infty$, there is typically no closed form unless it is an antiderivative. –  Ron Gordon Jan 10 '13 at 13:41
    
Still I would be interested where this integral comes from. –  Fabian Jan 10 '13 at 13:56
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1 Answer 1

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I found the solution from a reference paper, which is: A Double Integral Containing the Modified Bessel Function: Asymptotics and Computation.

http://www.ams.org/journals/mcom/1986-47-176/S0025-5718-1986-0856712-X/S0025-5718-1986-0856712-X.pdf

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Nice find, but where is the particular integral the OP wanted? –  Ron Gordon Apr 12 '13 at 13:05
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