Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
up vote 0 down vote accepted

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

share|cite|improve this answer
Nice find, but where is the particular integral the OP wanted? – Ron Gordon Apr 12 '13 at 13:05

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.