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Today I came across the integral

$\int_a^\infty e^{-bx} I_n(x) dx$

where $I_n$ is the modified Bessel function of the first kind. There is a solution for $a=0$, provided in Gradshteyn and Ryzhik, but I am afraid no closed-form solution exists otherwise. Correct me if I am wrong!

flag	asked Sep 28 2011 at 20:43 Xi'an 53●4

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Closed form in terms of which functions? – Yemon Choi Sep 28 2011 at 20:50

Obviously, this is a function of a I could call the modified Bessel function of the third kind! HOwever, I look for an expression I can program without creating a special code to bypass the integration... By the way, I was pointed out to the fact that b>1 is a necessary and sufficient condition for the integral to exist. Which is the case for my problem. – Xi'an Sep 30 2011 at 5:35

Igor Rivin · Accepted Answer · 2011-09-28 22:52:19Z UTC

I suspect your fears are justified. You are basically trying to integrate the integrand from $0$ to $a$ (since, as you have noted, and Mathematica confirms, the integral from $0$ has a closed form) You are thus trying to evaluate the indefinite integral of a Bessel function times an exponential, which does not exist in closed form (unless you call it a special function, and thus, presto, closed form, as @Yemon notes.)

A Bessel integral

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