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Topic: Integral
Replies: 5   Last Post: Jul 22, 2011 3:00 PM

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Rancid Moth

Posts: 505
Registered: 12/6/04
Integral
Posted: Jul 20, 2011 8:42 PM
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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?



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