Re: Fourier Help

• To: mathgroup at smc.vnet.net
• Subject: [mg43655] Re: Fourier Help
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Sun, 28 Sep 2003 06:00:35 -0400 (EDT)
• Organization: The University of Western Australia
• References: <bl3kht\$epr\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <bl3kht\$epr\$1 at smc.vnet.net>,
Arnold Gregory Civ AFRL/SNAT <Gregory.Arnold at wpafb.af.mil> wrote:

> I'm working with ver 5 & I've found a strange feature of the
> FourierTransform.  I was trying to reproduce the following transformation
> pair in Mathematica:
>
> FourierTransform[(1 - Sign[-1 + x1^2 + x2^2])/2,{x1,x2},{k1,k2}]=
> (2*Pi*BesselJ[1, Sqrt[k1^2 + k2^2]] ) / Sqrt[k1^2 + k2^2]
>
> Basically, this is a unit disk centered at the origin.  I've tried
> representing it as a unit step, too with no differences obtained.
> Mathematica 5 yields a strange mixed & incomplete (wrong?!?) result:
>
> {Sqrt[Pi/2]*DiracDelta[k1] -
>   Sqrt[Pi/2]*DiracDelta[k1]*Sign[-1 + x1^2 + x2^2],
>  Sqrt[Pi/2]*DiracDelta[k2] -
>   Sqrt[Pi/2]*DiracDelta[k2]*Sign[-1 + x1^2 + x2^2]}

I was not patient enough to see if Mathematica also returned this result.

> Notice that this is a list with 2 terms!?!  And a function of both the x's
> and k's?!?   The 1D version of this works (I didn't check it), but it didn't
> specifically return the bessel function.
>
> Mathematica  4.2 returned the original input with some error notations.
>
> Does anybody know of a more complete set of transform tables and / or a
> simple workaround. Obviously I could encode this particular transform
> directly, but if somebody else has already fixed this & other transforms I'm
> likely to need...

Clearly, a change of variables to polar coordinates is the right
approach here -- {x1 -> r Cos[q], x2 -> r Sin[q]} -- and this approach
is appropriate for a wide class of two-dimensional Fourier transforms.
The argument of the integral is then 0 for r > 1 and 1 for 0 <= r <= 1.
Hence the Fourier Transform is equivalent to computing the following
double integral:

Assuming[k > 0 && r > 0,
Integrate[r E^(I k r Cos[q]), {r, 0, 1}, {q, 0, 2 Pi}]]

which evaluates to

2 Pi BesselJ[1, k]/k

The order of integration is important (to Mathematica). It is better to
perform the q integration before the r integration.

It is probably not reasonable to expect Mathematica to perform the
change of variables or the simplification of sgn automatically.

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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