From: <andreyk1_at_post.tau.ac.il>

Date: Sat, 26 Aug 2006 02:28:30 +0300

This message was sent using IMP, the Internet Messaging Program. Received on Sat Aug 26 2006 - 01:28:42 MEST

Date: Sat, 26 Aug 2006 02:28:30 +0300

Hi Lorenzo and George,

The integral I have to calculate actually is not just J_0(x*y), but rather complicated, namely:

F(\alpha) = \int_{0}^{T} dx \int_{0}^{inf} dy BesselK0(\alpha*x*y)BesselJ0(x*y)

where T is a limit of integration over x and \alpha is a parameter.

I need to obtain a function squared of \alpha (F^{2}(\alpha)). Here starts a problem. Because integral squared actually is (\int F(x)dx)^{2} = \int F(x)dx\int F(y)dy, so, in order to obtain the right answer, I should use a triple integration, but this is not a problem. The problem is in convergence of integration. I use IntegralMultiple and can control the maximal number of poitns (iteration) - max_p, but increasing the nunmber (more than 1e+7) does not stabilize the result, it changes every time I change the number of max_p. Is there some sophisticated way to resolve the problem of integration of Bessel oscillating function?

Thanks for cooperating.

Quoting George Japaridze <george.japaridze_at_gmail.com>:

*> Hi,*

*>*

*> Lorenzo's right, the integral*

*>*

*> \int^{infinity}_{0} dx \int^{\infinity}_{0} dy J_{0}(x*y)*

*>*

*> is equal to one dimensional integral*

*>*

*> 2* \int^{\pi}_{0} dx/sin(x)*

*>*

*> which does not exists as an ordinary function - can be redefined as a*

*> generalized function (distribution).*

*> Depends on a specifics of the problem and are you or are you not*

*> allowed to use regularization.*

*>*

*> Cheers,*

*>*

*> George*

*> On Aug 25, 2006, at 9:50 AM, Lorenzo Moneta wrote:*

*>*

*> > Hi Andrew,*

*> >*

*> > actually I noticed now you want the integral between 0 and inf.*

*> > This is undefined, it is like getting the value of sin(x) for x=inf.*

*> >*

*> > Best Regards,*

*> > Lorenzo*

*> >*

*> >*

*> > On Aug 25, 2006, at 12:49 PM, andreyk1_at_post.tau.ac.il wrote:*

*> >*

*> >> Dear Lorenzo,*

*> >>*

*> >> Thanks for solution, but this is not my case. The integral is too*

*> >> complicated to*

*> >> be done analitically. The only way is numerical calculation.*

*> >> Actually I can*

*> >> integrate between BesselJ zeroes and sum up these intervals, but*

*> >> it is not*

*> >> efficient.*

*> >>*

*> >> Andrew.*

*> >>*

*> >> Quoting Lorenzo Moneta <Lorenzo.Moneta_at_cern.ch>:*

*> >>*

*> >>> Hi Andrew,*

*> >>>*

*> >>> I don't know of a method for 2d with oscillatory function.*

*> >>> Hower, you should be able for a Bessel of order 0 to solve the*

*> >>> integral analytically,*

*> >>> using the Bessel J0 definition*

*> >>>*

*> >>> see http://en.wikipedia.org/wiki/Bessel_function*

*> >>>*

*> >>> Cheers,*

*> >>>*

*> >>> Lorenzo*

*> >>> On 24 Aug 2006, at 12:14, andreyk1_at_post.tau.ac.il wrote:*

*> >>>*

*> >>>>*

*> >>>>*

*> >>>> Dear rooters,*

*> >>>>*

*> >>>> I need to integrate over two dimensionl oscillating function*

*> >>>> (TMath::BesselJ(0,x*y), 0<x<inf, 0<y<inf). I use TF1 with*

*> >>>> IntegralMultiple, but*

*> >>>> the answer is not stable, a change in max_points leads to a*

*> >>>> different result. I*

*> >>>> tried to increas the number of max_points to 1e+6 but the result is*

*> >>>> still*

*> >>>> unstable. Is there some method of integrating over oscillating*

*> >>>> function with*

*> >>>> arbitry dimensions? (in my case it is dim = 2).*

*> >>>>*

*> >>>>*

*> >>>> Thanks a lot!*

*> >>>>*

*> >>>>*

*> >>>> Andrew*

*> >>>>*

*> >>>> ----------------------------------------------------------------*

*> >>>> This message was sent using IMP, the Internet Messaging Program.*

*> >>>>*

*> >>>>*

*> >>>*

*> >>>*

*> >>> +++++++++++++++++++++++++++++++++++++++++++*

*> >>> This Mail Was Scanned By Mail-seCure System*

*> >>> at the Tel-Aviv University CC.*

*> >>>*

*> >>*

*> >>*

*> >>*

*> >>*

*> >> ----------------------------------------------------------------*

*> >> This message was sent using IMP, the Internet Messaging Program.*

*> >>*

*> >*

*>*

*>*

*>*

*>*

*>*

*>*

*>*

*>*

*>*

*>*

*>*

*>*

*> *************************************************************************

*> ********************

*> Dr. George Japaridze CTSPS, Clark*

*> Atlanta University*

*>*

*> japar_at_ctsps.cau.edu 404 880 6420*

*> Office*

*> http://www.robotics.cau.edu/people/japar404 226 3847 Cell*

*>*

*>*

*>*

*>*

*>*

*>*

This message was sent using IMP, the Internet Messaging Program. Received on Sat Aug 26 2006 - 01:28:42 MEST

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