A Bessel integral
- To: mathgroup at smc.vnet.net
- Subject: [mg36779] A Bessel integral
- From: Roberto Brambilla <rlbrambilla at cesi.it>
- Date: Wed, 25 Sep 2002 01:50:58 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Hi to all Mathematica friend!
I am considering the following integral
W[m_,n_]:=Integrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity}]
where m,n are reals >=0. With Mathematica 4.1 I obtain:
If[Re[m+n]>-1, -Cos[(m-n)Pi/2]/(2 Pi)*
(2 EulerGamma + Log[4] +
PolyGamma[0, 1/2(1 + m - n)] +
PolyGamma[0, 1/2(1 - m + n)] +
2PolyGamma[0, 1/2(1 + m + n)])
and so using this answer as a definition I obtain
W[0,0]=-(2 EulerGamma + Log[4] + 4 PolyGamma[0, 1/2])/(2 Pi)=0.84564
I suspect that these integrals are divergent (*). So I try the numerical
integration:
NW[m_,n_]:=NIntegrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity}]
so that
NW[0,0]=11.167
Othe couples are
W[1,0]=Indeterminate NW[1,0]=0.597973
W[0,1.5]=0.537095 NW[0,1.5]=-5.79306
W[1,1]=0.20902 NW[1,1]=17.5425
W[2,0]=0.427599 NW[2,0]=-6.83464
W[2,1]=Indeterminate NW[2,1]=4.69013
(*) The integral is a particular case of the Weber-Schafheitlin integrals
(Abramowitz, 11.4.33).
Any explanation about the analytical expression will be gratefully accepteed.
Roberto.
Roberto Brambilla
CESI
Via Rubattino 54
20134 Milano
tel +39.02.2125.5875
fax +39.02.2125.5492
rlbrambilla at cesi.it