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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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