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Topic: Int(exp*BesselI0(sqrt)): Gradshteyn & Ryshik, 6.616.5 ?
Replies: 5   Last Post: Oct 6, 2013 12:42 PM

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G. A. Edgar

Posts: 2,499
Registered: 12/8/04
Re: Int(exp*BesselI0(sqrt)): Gradshteyn & Ryshik, 6.616.5 ?
Posted: Oct 5, 2013 9:07 AM
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In article <524FF9CF.6F7CD843@freenet.de>, <clicliclic@freenet.de>
wrote:

> Axel Vogt schrieb:
> >
> > The book says: Int( exp(-x*a) * BesselI(0, b*sqrt(-x^2+1)), x = -1 .. 1) =
> > = 2*sinh(a^2+b^2)/sqrt(a^2+b^2)
> >
> > For a=0 I obtain that it is 2*sinh(b)/b, using Maple (sketch: write Bessel
> > as series and integrate termwise; evaluating the infinite series gives it).
> > Numerical tests do confirm that.
> >
> > Is somebody aware of the root of the proof in G&R (can not find an erratum,
> > but some usage of the formula) to eliminate the possible typo?

>
> There is no entry 6.616.5 for
>
> INT(exp(-x*a)*BesselI(0, b*sqrt(1-x^2)), x, -1, 1)
>
> in my 1981 edition of Gradshteyn & Ryzhik; it must have been added
> later. I suggest to try the modified evaluation
>
> 2*sinh(sqrt(a^2+b^2))/sqrt(a^2+b^2)
>
> which holds at b=0 as well as a=0, since BesselI(0, 0) = 1. It is also
> in full agreement with a numerical evaluation of the integral on Derive
> giving 2.199018052 for a = 0.3 and b = 0.7. Doing the general integral
> symbolically may be difficult; related integrals under 6.616 in my
> edition are all taken from Magnus and Oberhettinger, 1948.
>
> You could file an error report with the G&R editors (after checking the
> latest edition).
>
> Martin.


In my 1980 copy of G&R (English translation of the fourth Russian
edition), there is such an entry, with note 3 meaning "added in the
third edition". However (as Axel suggests) the value is
2*sinh(sqrt(a^2+b^2))/sqrt(a^2+b^2)

--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/



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