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MathGroup Archive 2004

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Re: integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46603] Re: integral
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 25 Feb 2004 13:06:58 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <c1ehlu$kfu$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <c1ehlu$kfu$1 at smc.vnet.net>,
 Fadaa Nicolas <fadaa.nicolas at coria.fr> wrote:

> I need to evaluate the the following integral:
> 
> Integrate[Exp[-a^2*(x-x0)^2] BesselJ[1,bx] Sin[xY]}]
> x0, b and y are fixed
> 
> Does anyone know, if there is a primitive to it ?

First note that you need to include a space between the b and x and x 
and Y. Secondly, capital letter variables are generally not a good idea 
(especially C, D, E, I).

Mathematica can compute the following integral:

  int[a_,x0_][y_] = Integrate[x Exp[-a^2 (x-x0)^2] Sin[x y], x]

Since BesselJ[1,z] is equivalent to

  z/2 Hypergeometric0F1[2, -z^2/4] // FunctionExpand

the general term in the series expansion of BesselJ[1,z] is

  z/2 (-z^2/4)^n/n!^2/(n + 1),

and expansion in odd powers of z. Furthermore, since

  D[Sin[x y], {y, 2}]

is

  (-x^2) Sin[x y]

you can obtain the integral you want as an infinite sum of (parametric) 
derivatives with respect to y of int[a,x0][y].

However, even the basic integral is rather complicated, and computing 
all the required derivatives is unlikely to simplify things.

Instead, if you want to compute this integral over a specified domain 
then using NDSolve (instead of NIntegrate) is probably the best approach:

 With[{a=1, x0=2, b=3, y=1},
   NDSolve[{f[0]==0,f'[x]==Exp[-a^2 (x-x0)^2] BesselJ[1, b x] Sin[x y]},
     f,{x,0,16}]]

This returns the antiderivative as an InterpolatingFunction. 

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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