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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
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