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Re: A Bessel integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg36848] Re: [mg36779] A Bessel integral
*From*: Vladimir Bondarenko <vvb at mail.strace.net>
*Date*: Sun, 29 Sep 2002 02:55:09 -0400 (EDT)
*Reply-to*: Vladimir Bondarenko <vvb at mail.strace.net>
*Sender*: owner-wri-mathgroup at wolfram.com
Roberto Brambilla <rlbrambilla at cesi.it> wrote on Wed, 25 Sep 2002 01:50:58 :
RB> I am considering the following integral
RB> W[m_,n_]:=Integrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity}]
RB> where m,n are reals >=0. With Mathematica 4.1 I obtain:
RB> If[Re[m+n]>-1, -Cos[(m-n)Pi/2]/(2 Pi)*
RB> (2 EulerGamma + Log[4] +
RB> PolyGamma[0, 1/2(1 + m - n)] +
RB> PolyGamma[0, 1/2(1 - m + n)] +
RB> 2PolyGamma[0, 1/2(1 + m + n)])
RB> Any explanation about the analytical expression will be
RB> gratefully accepteed.
The expression for W[m_,n_] returned by Mathematica is wrong.
To prove, just substitute m = n = 0 which is exactly what you had done
and observe that the output you had had
W[0,0]=-(2 EulerGamma + Log[4] + 4 PolyGamma[0, 1/2])/(2 Pi)
= 0.84564
was incorrect. The correct answer is 1/2.
Mathematica can handle the numeric integration successfully
In[1] := NIntegrate[BesselJ[1, x]*BesselJ[0, x], {x, 0, Infinity},
Method -> Oscillatory]
(* The warnings are skipped *)
Out[1] = 0.5
Using NIntegrate[BesselJ[0, x]*BesselJ[0, x], {x, 0, Infinity}]
without Method -> Oscillatory is not the optimal choice as
the integrand oscillates fairly rapidly over the integration
region.
RB> I suspect that these integrals are divergent (*).
In fact, not exactly.
Integrate[BesselJ[1, x]*BesselJ[0, x], {x, 0, Infinity}]
is equal to 1/2, and Mathematica 4.1 for Microsoft Windows
(November 2, 2000) does it correctly, while Mathematica 4.2
for Microsoft Windows (February 28, 2002) concocts a strange
mixture of a wrong divergence message and the warning that
it cannot check the convergence [should I trust to the second
warning? or the first?]
As a matter of fact,
Integrate[BesselJ[1, x]*BesselJ[0, x], {x, 0, Infinity}]
converges because the integrand is regular at x=0, bounded over
the whole right semi-axis, and decays as
2*Cos[Pi/4 - x]*Cos[(3*Pi)/4 - x]/(Pi*x) + o(1/x)
at x -> Infinity .
Say, calculate
Normal[Series[BesselJ[1, x], {x, Infinity, 1}]] Normal[
Series[BesselJ[0, x], {x, Infinity, 1}]] // InputForm
->
(2*(Cos[Pi/4 - x] - Sin[Pi/4 - x]/(8*x))*(Cos[(3*Pi)/4 - x] +
(3*Sin[(3*Pi)/4 - x])/(8*x)))/(Pi*x)
then Plot[%,{x,1,10}]
and Plot[BesselJ[1,x]*BesselJ[0,x],{x,1,10}]
and you could hardly see the difference.
Generally, to get to the convergence domain for W in terms of
m and n is easy via the asymtotics of the Bessel functions
(use something like
Expand[Normal[Series[BesselJ[m, x], {x, Infinity, 1}]]Normal[
Series[BesselJ[n, x], {x, Infinity, 1}]]]
then analyze the main term).
Best wishes,
Vladimir Bondarenko
Mathematical Director
Symbolic Testing Group
Email: vvb at mail.strace.net
Web : http://www.CAS-testing.org/
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