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Topic: BesselI integral identity
Replies: 6   Last Post: Mar 2, 2007 3:09 AM

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Paul Abbott

Posts: 1,437
Registered: 12/7/04
Re: BesselI integral identity
Posted: Feb 26, 2007 10:16 PM
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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.ohio-state.edu.invalid>
> wrote:

> > In article <paul-D840B2.23254223022...@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 well-known, 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 ...


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

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