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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
Re: Integral
Posted: Jul 22, 2011 7:28 AM
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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 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.



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