
Re: BesselI integral identity
Posted:
Feb 26, 2007 10:16 PM


In article <1172264479.147458.90040@t69g2000cwt.googlegroups.com>, "Robert Israel" <israel@math.ubc.ca> wrote:
> On Feb 23, 11:50 am, "G. A. Edgar" <e...@math.ohiostate.edu.invalid> > wrote: > > In article <paulD840B2.23254223022...@news.uwa.edu.au>, Paul Abbott > > > > <p...@physics.uwa.edu.au> wrote: > > > I have managed to show that (in Mathematica notation) > > > > > Integrate[Exp[a x] BesselI[0, b Sqrt[1  x^2]], {x, 1, 1}] == > > > 2 Sinh[Sqrt[a^2 + b^2]] / Sqrt[a^2 + b^2] > > > > > I expect that this identity is wellknown, but I cannot find it in > > > Abramowitz and Stegun, nor can I obtain it (directly) from other > > > identities therein. Can anyone point me to a reference where this > > > identity, or its generalization, is obtained? > > > > Gradshteyn & Ryzhik, 6.616.5 is exactly this, and has the assumptions > > a>0, b>0.
Thanks for this. I should have looked there  especially since I have a copy! I also looked in
Tables of Summable Series and Integrals Involving Bessel Functions
by Albert D. Wheelon, but could not find it there.
> However, those assumptions are not necessary, since both sides are > invariant under sign changes of a or b.
A good point. Also, at least in the fifth edition of GR, no reference is cited for this formula. However, there is a superscript 3 above the 5 for formula 6.616.5, yet I can find no explanation of what that superscript means. I assume it is a footnote/endnote reference ...
Cheers, Paul
_______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul

