I was faced with the same situation, it seemed to me. The difference is, I knew what the closed form should look (from checking A&S), and I was stuck in Mathematica (MMA), surprisingly. It turned out that MMA could get there, thankfully, but it needed to be pointed in the right direction by telling it the appropriate range for the various constants.

My thought was that perhaps MMA or similar tools could also resolve your problem. You didn’t mention if you tried that and I didn’t notice it mentioned by other commentators.

]]>I had to check my copy of A&S (15.1.20) to (i) verify the identity, and (ii) note the range of coefficient values for which it applies. Using FunctionExpand, with those A&S ranges incorporated in an Assumptions qualifier, did the trick. The apparent intransigence of MMA here is a feature, not a bug. MMA is so general that it just needed to be nudged in the desired direction.

BTW, A&S is also available online. In a pinch, the familiar is more valuable than the new. :)

]]>http://dlmf.nist.gov/10.30 (Hint: Use the DLMF; it’s the official successor to Abramowitz & Stegun and the nice approach of being on the web (and available as nice book+ CD-rom)

Actually it follows from that the integral is finite exactly when b > 1.

]]>I’m pretty sure you could prove that the integral is divergent for

0 < b <= 1. ]]>

intBesselI <- function(a,b,n) integrate(function(x) exp(-b*x)*besselI(x,nu=n),lower=a, upper =1000)

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