Re: Strange results from Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg75017] Re: Strange results from Mathematica*From*: "dimitris" <dimmechan at yahoo.com>*Date*: Sat, 14 Apr 2007 01:09:00 -0400 (EDT)*References*: <evn8cr$s7$1@smc.vnet.net>

$VersionNumber 5=2E2 The integral (1) is divergent and Mathematica 5.2 understands this. At zero we have: x*BesselJ[0, x]^2 + O[x, 0]^6 SeriesData[x, 0, {1, 0, -1/2, 0, 3/32}, 1, 6, 1] so the integrand gives no trouble there. The only problematic location is infinity. Limit[x*BesselJ[0, x]^2, x -> Infinity] Interval[{-(1/Pi), 3/Pi}] Integrate[x*BesselJ[0, x]^2, {x, 0, Infinity}] Integrate::gener: Unable to check convergence. Integrate::idiv: Integral of x*BesselJ[0, x]^2 does not converge on \ {0,Infinity}. Integrate[x*BesselJ[0, x]^2, {x, 0, Infinity}] Try this to your Mathematica to see what you get (you should get Infinity) (*Ins*) Integrate[x*BesselJ[0, x]^2, x] Limit[%, x -> Infinity] - Limit[%, x -> 0] (*Outs*) (1/2)*x^2*(BesselJ[0, x]^2 + BesselJ[1, x]^2) Infinity (*That is, evaluate first the indefinite integral and then applying the Newton-Leibniz formula to the obtained antiderivative.) As regards the second integral we have Integrate[BesselJ[0, x]^2, {x, 0, Infinity}] Integrate[BesselJ[0, x]^2, {x, 0, Infinity}] There are three possible explanations 1) There is not closed form solution or 2) There is closed form solution and Mathematica does not know it or 3) There is a closed form solution which can't be expressed in terms of the implemented built in functions of Mathematica. Also Integrate[BesselJ[0, x]^2, x] Limit[%, x -> Infinity] - Limit[%, x -> 0] x*HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -x^2] Limit[x*HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -x^2], x -> Infinity] Mathematica is not able to get the last limit BUT x*HypergeometricPFQ[{1/2, 1/2}, {1, 1, 3/2}, -x^2] + O[x]^6 SeriesData[x, 0, {1, 0, -1/6, 0, 3/160}, 1, 6, 1] So I think the integral is actually divergent. Below we get some other arguments for this statement. Integrate[x^n*BesselJ[0, x]^2, {x, 0, Infinity}, Assumptions -> -1 < Re[n] < 0] Limit[%, n -> 0] (Gamma[-(n/2)]*Gamma[(1 + n)/2])/(2*Sqrt[Pi]*Gamma[1/2 - n/2]^2) -Infinity Integrate[Exp[(-x)*s]*BesselJ[0, x]^2, {x, 0, Infinity}, Assumptions - > Re[s] > 0] Limit[%, s -> 0] (2*EllipticK[-(4/s^2)])/(Pi*s) Infinity =CF/=C7 Jung-Tsung Shen =DD=E3=F1=E1=F8=E5: > Hi, I encountered a rather strange results from Mathematica today. > > When evaluating the following integration > > (1) Integrate[x (BesselJ[0, x])^2, {x,0, infinity}] > > my Mathematica gave the answer 0. That's very weird, since if you plot > out the function x (BesselJ[0, x])^2, apparently the integral should > not be zero ... > > Moreover, for the following integration > > (2) Integrate[ (BesselJ[0, x])^2, {x,0, infinity}] > > My Mathematica says it's some finite value. However, once you did > this, and then did (1), and then came back to do (1), the answer > became 0 ... > > My Mathematica version is 5.0.1.0 on Mac Powerbook. > > Thanks. > > JT