```Date: Jul 22, 2011 7:28 AM
Author: Rancid Moth
Subject: Re: Integral

On Thu, 21 Jul 2011 16:53:20 -0700, Dann Corbit <dcorbit@connx.com>wrote:>In article <3ise275f397jphilkph0rpnhb99k3gv46l@4ax.com>, >rancidmoth@yahoo.com says...>> >> Hello all,>> >> I'm trying to evaluate the following integral>> >> S(a,b) = integrate(0,2pi) sin(a*cos(t))*sin(b*sin(t)) />> (sin(t)*cos(t)) dt>> >> I get S(a,b) = 16pi*a*b*J(2,sqrt(a^2+b^2))/(a^2+b^2)>> >> where J(k,z) is the bessel function of the first kind, order k.>> However funny numerics started occuring and i think i have traced it>> back to this integral.  It would appear (in mathematica and maple)>> that numerically this appears reasonable for a,b<1.  but for a,b>1 the>> numerical integration and my result diverge quite significantly...too>> much to perhaps be numerical error in the integration.  >> >> My steps are as follows:>> >> cos(a*z+b/z) = -J(0,2sqrt(ab)) + sum(k=0,oo) (-1)^k ((a*z)^(2k) +>> (b/z)^(2k))*J(2k,2sqrt(ab))/(sqrt(ab))^(2k)>> >> I derived this by using binomial theorem in the series for cos - also>> numerically it appears bang on.  Using this, convering the integral to>> that over the unit circle, yeilds my result.>> >> Alternatively one may use similar laurent series for sin(z+1/z),>> multiply them together, get the residue and get the same result.>> >> what have i missed?>>http://integrals.wolfram.com/index.jsp?expr=sin%28a*cos%28t%29%29*sin%>28b*sin%28t%29%29+%2F%28sin%28t%29*cos%28t%29%29+&random=false>>result:>x Csc[t] Sec[t] Sin[a Cos[t]] Sin[b Sin[t]]Yes that result is obvious...however you want to integrate w.r.t to_t_, not the default _x_ which is what the wolfram integrator does.However the more i think about it the more i believe thatmathematica/maple's integration numerical error is indeed the issue. Ithink i can use the Riemann-Lebesgue lemma to show that as either a orb -->oo then the integral should go to zero.  this is indeed whatoccurs with my bessel function result...but _not_ with the numericalresults in either package.  in either maple or mathematica, for verylarge a or b, the numerical integration yeilds a large and largernumber, until eventually a,b, become so large the numerical algorithmfails to converge.
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