Re: Oscillating method of integration

From: <andreyk1_at_post.tau.ac.il>
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
> >>>>
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> >>>>
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> >>>
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> >>>
> >>
> >>
> >>
> >>
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>
>
>
>
>
>
>
>
>
>
>
>
> ************************************************************************
> *******************
> 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
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This message was sent using IMP, the Internet Messaging Program. Received on Sat Aug 26 2006 - 01:28:42 MEST