Date: Jul 22, 2011 7:28 AM
Author: Rancid Moth
Subject: Re: Integral
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.