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Re: Integral no longer evaluated in Version 7, 8

  • To: mathgroup at smc.vnet.net
  • Subject: [mg114738] Re: Integral no longer evaluated in Version 7, 8
  • From: John Jowett <john.m.jowett at gmail.com>
  • Date: Tue, 14 Dec 2010 06:58:18 -0500 (EST)
  • References: <idd7oq$ncm$1@smc.vnet.net>

Thanks Daniel,
         Meanwhile I found a solution based on splitting the range of
integration and a simple change of variable:

f1 = Integrate[(1/2)*x^2*BesselK[5/3, x], {x, 0, 1}]

f2 = Integrate[(BesselK[5/3, 1/y]/(2*y^2))*(-(y^2)^(-1)), {y, 1, 0}]

f1 + f2 // FullSimplify

which gives the correct result

(8 \[Pi])/(9 Sqrt[3])

I'm a little surprised that Mathematica didn't try something similar.

(By the way, my original post should have said that Mathematica solved
this integral by itself in Version 6, not 7.)

John Jowett

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[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 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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