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

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Axel Vogt

Posts: 1,068
Registered: 5/5/07
Re: Integral
Posted: Jul 22, 2011 10:01 AM
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On 21.07.2011 02:42, wrote:
> 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?

For a=1/2/2, b=1/2 I get 0.765149580251623 for the integral and
0.765143786388934 for your formula, using Maple with 15 Digits.
So it seems that it is not correct for a,b<1 as well.

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