Re: Integral no longer evaluated in Version 7, 8

*To*: mathgroup at smc.vnet.net*Subject*: [mg114430] Re: Integral no longer evaluated in Version 7, 8*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 4 Dec 2010 06:13:27 -0500 (EST)

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