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)
• 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

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,

Mathematical Director
Symbolic Testing Group
Email:  vvb at mail.strace.net

Web  :  http://www.CAS-testing.org/

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Voice:  (380)-652-447325 Mon-Fri 9 a.m. - 6 p.m.
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```

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