On Thu, 21 Jul 2011 16:53:20 -0700, Dann Corbit <firstname.lastname@example.org> wrote:
>In article <email@example.com>, >firstname.lastname@example.org 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 that mathematica/maple's integration numerical error is indeed the issue. I think i can use the Riemann-Lebesgue lemma to show that as either a or b -->oo then the integral should go to zero. this is indeed what occurs with my bessel function result...but _not_ with the numerical results in either package. in either maple or mathematica, for very large a or b, the numerical integration yeilds a large and larger number, until eventually a,b, become so large the numerical algorithm fails to converge.