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[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 -