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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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