# Double Integral involving modifed bessel function

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