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Integral
Posted:
Jul 20, 2011 8:42 PM


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?



