## Bessel integral

Pierre Pudlo and I worked this morning on a distribution related to philogenic philogenetic trees and got stuck on the following Bessel integral

$\int_a^\infty e^{-bt}\,I_n(t)\,\text{d}t\qquad a,b>0$

where In is the modified Bessel function of the first kind. We could not find better than formula 6.611(4) in Gradshteyn and Ryzhik. which is for a=0… Anyone in for a closed form formula, even involving special functions?

### 14 Responses to “Bessel integral”

1. I didn’t look at your problem (sorry, no time) but you say you are “stuck” looking for a closed form for your integral, “even involving special functions.”

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.

2. Thanks, Neil, but I miss the connection with my own problem… Could you expand, please?

3. An ironic post because, at essentially the same time, I was battling with an integral from a Dirichlet distribution. MMA produced a solution expressed in terms of Hypergeometric2F1, which I knew should be further reducible to Gammas, but it wasn’t a happening thing, despite fiddling around with integration limits, etc.

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. :)

4. Hmm, but you really need b > 1, don’t you?
I’m pretty sure you could prove that the integral is divergent for
0 < b <= 1.

• Yes, that’s easy to prove from this formula

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.

• Thank you, Martin, my b is indeed larger than 1.

• And thanks for the DLMF reference too! Quite useful.

5. I suppose the naive numerical solution has problems for large a…

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

6. Interesting! I’d like to know more about this work. How/why does this nasty integral pop up? And I think you mean “phylogenetic” trees. Not “philogenic”

Cheers,

Simon

• Top secret of course!!!

7. Natesh Pillai Says:

you may try posting in

• Thanks, just did it!

8. Mathematica can’t do it it for general a, although can for a=0. I suspect this means there isn’t a nice answer.

• Thanks, Wolfram alpha could not do it either…. So here goes my hope for a nice and quick solution!

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