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MathGroup Archive 2002

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A Bessel Integral Re:

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36886] A Bessel Integral Re:[mg36848]
  • From: Roberto Brambilla <rlbrambilla at cesi.it>
  • Date: Tue, 1 Oct 2002 04:45:06 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

Thank you Vladimir
for your extensive answer but 
I still have some doubts about convergence of the following integral (m,n
integrers>=0)

W[m_,n_]:=Integrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity}]

for wich Mathematica gives the close form 

W[m_,n_]:= -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)] )


You say this integral is convergent to 1/2 for m=0 and n=1.
Also Mathematica agrees to you since for m>=0

W[m,m+1]=1/2
W[m,m+3]=-1/2

Numerically we have

NIntegrate[BesselJ[0, x]*BesselJ[1, x], {x, 0, Infinity}]
NIntegrate::"ncvb": "NIntegrate failed to converge to prescribed accuracy....
0.597973

NIntegrate[BesselJ[0, x]*BesselJ[1, x], {x, 0, Infinity}, Method ->
Oscillatory]
NIntegrate::"ploss" : ....
0.5
So I define also the corresponding  numeric definition

NW[m_, n_] := NIntegrate[BesselJ[m, x]*BesselJ[n, x], {x, 0, Infinity},
Method -> Oscillatory]


THEORY
The integral is the critical case of Weber-Schafheitlin integral
(see Watson book on Bessel function p.402, or Ryzhik-Gradshteyn 6.574(2)).
According to this theory

WS[m_,n_,p_]:=Integrate[BesselJ[m, x]*BesselJ[n, x] x^-p, {x, 0, Infinity}]
= A/B

where

A=Gamma[p]*Gamma[(n+m-p+1)/2]
B=2^p Gamma[(n-m+p+1)/2]Gamma[(n+m+p+1)/2]Gamma[(m-n+p+1)/2]

By the presence of Gamma[p] in numerator A, in the case p=0 as in W[m,n]
all these integrals are divergent since Gamma[0]=Infinity.
The integral exist if m+n+1 > p > 0.

ASYMPTOTICS
The Watson theory is in conflict with Mathematica and your notes according
which
the asyntotic trend 1/x of the integrand in W[m,n] is enough for
convergernce. 
I divide the integral in two parts

Wasy[m_,n_,a_]=NIntegrate[BesselJ[0, x]*BesselJ[1, x], {x, 0, a]+
               NIntegrate[BesselJ[0, x]*BesselJ[1, x], {x, a, Infinity}]

and if a>>1 I use asyntotic expansion of Bessel function in the second
integral
so that  I can write

Wasy[m_,n_,a_]= int1[m,n,a]+int2[m,n,a]

where

int1[m_,n_,a_]:=NIntegrate[BesselJ[0, x]*BesselJ[1, x], {x, 0, a]+
int2[m_,n_,a_]:=(2/Pi)Integrate[Cos[x-(2m+1)Pi/4]*Cos[x-(2n+1)Pi/4], {x, a,
Infinity}]

The first integral is a quite normal finite integral. The second (int2) is
singular
and according to Mathematica 4.1 

int2[m_, n_, a_] := -(1/Pi)*Log[a]*Cos[1/2(m - n)Pi]*]Log[a] + 
                     (1/Pi)*CosIntegral[2 a]*Sin[1/2(m+n)Pi] + 
                      1/(2*Pi)*Cos[1/2(m+n)Pi]*(Pi-SinIntegral[2*a])
                 
RESULTS

m=1;n=0;a=20.;
WS[m,n,0]=divergent
W[m,n]=1/2
NW[m,n]=0.5
Wasy[m,n,a]=.49816

m=2;n=0;a=20.;
WS[m,n,0]=divergent
W[m,n]=0.427599
NW[m,n]=-2.43818
Wasy[m,n,a]=-1.48052

m=3;n=1;a=20.;
WS[m,n,0]=divergent
W[m,n]=0.639806
NW[m,n]=-2.31957
Wasy[m,n,a]=-1.26822

m=4;n=0;a=20.;
WS[m,n,0]=divergent
W[m,n]=-.852012
NW[m,n]=1.45786
Wasy[m,n,a]=1.06835


The cases W[m,m+1],W[m,m+3] well agrre with the numerical counterpart.
Other case are doubtfully.
I think the main problem is the convergence of this kind of integrals.
Any suggestion will be well considerd.

Robert






Roberto Brambilla
CESI
Via Rubattino 54
20134 Milano
tel +39.02.2125.5875
fax +39.02.2125.5492
rlbrambilla at cesi.it



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