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Re: Integral no longer evaluated in Version 7, 8

John Jowett wrote:
> Hello,
>         With Mathematica Version 7, the integral
> Integrate[(x^2/2)*BesselK[5/3, x], {x, 0, Infinity}]
> correctly evaluated to (8*Pi)/(9*Sqrt[3]).  In Mathematica 7 or 8, it
> gives the message
> Integrate::idiv: Integral of x^2 BesselK[5/3,x] does not converge on
> {0,\[Infinity]}. >>
> I haven't been able to find any way to get this to work (NIntegrate
> works fine). Termwise integration of the asymptotic form of the
> integrand works but does not appear to converge.
> Can anybody explain why Mathematica lost this capability?  It may have
> something to do with no longer recognising cancellations among
> expressions involving the Gamma function.  Any ideas for getting the
> integral to work ?
> Thanks,
> John Jowett

It's a known bug, caused by a bad series expansion at infinity for the 
antiderivative of that integrand.

i1 = (x^2/2)*BesselK[5/3, x];
i2 = Integrate[i1, x];
i3 = Normal[Series[i2, {x, Infinity, 3}]];

Now compare plots (the first is to show that it very likely is 
convergent based on integrand behavior).

Plot[i1, {x, 2, 20}]
Plot[i2, {x, 2, 20}]
Plot[i3, {x, 2, 20}]

Daniel Lichtblau
Wolfram Research

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