       Re: Integral no longer evaluated in Version 7, 8

• To: mathgroup at smc.vnet.net
• Subject: [mg114739] Re: Integral no longer evaluated in Version 7, 8
• From: John Jowett <john.m.jowett at gmail.com>
• Date: Tue, 14 Dec 2010 06:58:29 -0500 (EST)
• References: <idd7oq\$ncm\$1@smc.vnet.net>

```Further to my last post, it isn't even necessary to split the range, a
simple change of variables x->1/y does the trick:

Integrate[(BesselK[5/3, 1/y]/(2*y^2))*(-(y^2)^(-1)), {y, Infinity, 0}]

and I'm even more surprised that Mathematica didn't try it ,,,

John

On Dec 4, 12:13 pm, Daniel Lichtblau <d... at wolfram.com> wrote:
> 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).  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 hav=
e
> > 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- Hide quoted text -
>
> - Show quoted text -

```

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