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I have the following double integral: $\int\limits_0^x {\int\limits_0^y {{e^{ - {K_1}(u + v)}}{I_0}\left( {2{K_1}\sqrt {uv} } \right)dudv} }$ where $K_1$ is a constant. Do you have any ideas of getting a closed form for this integral? Thank you very much.

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  • $\begingroup$ What is the function $I_0$? You can get rid of the constant by substitution. $\endgroup$ Commented Jan 10, 2013 at 9:36

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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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To Jochen Wengenroth, my guess is I_{0} is a modified Bessel function. To the OP. I have no expertise when it comes to this, but if it is a Bessel function, converting it to some polar form of some kind might help.

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    $\begingroup$ I found the solution from a reference paper, which is: A Double Integral Containing the Modified Bessel Function: Asymptotics and Computation. $\endgroup$
    – BinhDDT
    Commented Jan 23, 2013 at 11:34
  • $\begingroup$ For anyone interested, here's a link to the paper unknown (google) cited: ams.org/journals/mcom/1986-47-176/S0025-5718-1986-0856712-X/… $\endgroup$ Commented Jan 24, 2013 at 19:28
  • $\begingroup$ @Binh Nguyen: Please, post this as an answer and accept it, so that the software does not bump the question periodically thinking it is unanswered. $\endgroup$ Commented Feb 7, 2013 at 11:39

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