       Re: Fourier transform in arbitrary dimension?

• To: mathgroup at smc.vnet.net
• Subject: [mg88079] Re: Fourier transform in arbitrary dimension?
• From: Jens-Peer Kuska <kuska at informatik.uni-leipzig.de>
• Date: Wed, 23 Apr 2008 06:05:54 -0400 (EDT)
• Organization: Uni Leipzig
• References: <fumr2o$sh2$1@smc.vnet.net>

Hi,

integrateDimD[f_, dim_Integer] :=
Module[{var},
var = Table[{Unique[x], -Infinity, Infinity}, {dim}];
Integrate @@ {f @@ (First /@ var), Sequence @@ var}
]

and

integrateDimD[f, 3]

gives

Integrate[f[x$12006, x$12007, x$12008], {x$12006, -Infinity, Infinity},
{x$12007, -Infinity, Infinity}, {x$12008, -Infinity, Infinity}]

Regards
Jens

Barrow wrote:
> Dear all,
>
> I would like to calculate a Fourier transform in arbitrary dimension
> , say D, of the function 1/q^2, where q denotes the absolute value
> of a D dimensional spatial vector.
> The integral I have to perform is
>
> \int \frac{d^Dq}{(2\pi)^D}\exp(-iQ\cdot x)\frac{1}{q^2}
>
> where |Q| = q.
> But I can't find a way to tell Mathematica to calculate this integral
> "of dimension D."
> PS. The answer is proportional to \Gamma(D/2 - 1)(x^2/4)^{1-D/2}
>
> Any ideas would be appreciated.
> Sincerely Barrow
>



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