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