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MathGroup Archive 2002

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a difficult Bessel intergral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg32479] a difficult Bessel intergral
  • From: Roberto Brambilla <rlbrambilla at cesi.it>
  • Date: Wed, 23 Jan 2002 00:59:53 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Hi to all MG friend!

Solving an integral equation in polar coordinates I need
the following integral (that Mathematica4.1 cannot symb. solve)

L[x_,y_,s_] := Integrate[ p(p^2+s^2)^(-1/2) BesselJ[0, x p] BesselJ[0, y p],
               {p,0,Infinity}]

where x,y s are real numbers >=0. Special cases:

L[0,y_,s_] := Exp[-y s]/y
L[x_,0,s_] := Exp[-x s]/x
L[x_,y_,0] := (z=Min[x,y];Z=Max[x,y]; (2/Pi) EllipticK[(z/Z)^2]/Z )  

It looks like a common and innocent integral but tables I have (Abramowitz,
Gradshtein-Ryzhik)
don't consider it. I'd like to know if it is present in the more extensive
tables
(like Erdély, Prudnikov, Luke...) that I can't access.

Also the numerical computation gives a lot of troubles and is very slow
since the integrand is oscillatory. The option Method->(some method) does not
improve the evaluation. May be some anlytical prework has to be done before
numbers.
Any suggestion 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



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