Bug 2?

• To: mathgroup at smc.vnet.net
• Subject: [mg43711] Bug 2?
• From: Konstantin L Kouptsov <kouptsov at wsu.edu>
• Date: Tue, 30 Sep 2003 16:43:19 -0400 (EDT)
• Organization: Washington State University
• References: <bl3kht\$epr\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In[1]:= Timing[FourierTransform[1 - Sign[-1 + x^2 + (2 y)^2], {x, y}, {p, q}]]
Out[1]:= {64.4*Second,
{Sqrt[2*Pi]*DiracDelta[p] - Sqrt[2*Pi]*DiracDelta[p]*Sign[-1 + x^2 + 4*y^2],
Sqrt[2*Pi]*DiracDelta[q] - Sqrt[2*Pi]*DiracDelta[q]*Sign[-1 + x^2 + 4*y^2]}}

Knowing that
In[2]:= FourierTransform[1, {x}, {p}]
Out[2]:= Sqrt[2*Pi]*DiracDelta[p]

and in 2d

In[3]:= FourierTransform[1, {x, y}, {p, q}]
Out[3]:= 2*Pi*DiracDelta[p]*DiracDelta[q]

one notices that Mathematica dows not recognize x and y in the first example as
variables, and just takes the whole function (1 - Sign[-1 + x^2 + (2 y)^2]) as a
constant multiplied by a 1-d fourier of a 1 ( = Sqrt[2*Pi]*DiracDelta[p]), i.e.

FourierTransform[1 - Sign[-1 + x^2 + (2 y)^2], {x, y}, {p, q}]]
-> (1 - Sign[-1 + x^2 + (2 y)^2])*
{FourierTransform[1, {x}, {p}],FourierTransform[1, {x}, {q}]}
-> Nonsense result.

Arnold Gregory Civ AFRL/SNAT 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]}
>
> 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...
>
> Thanks for any help & insight you can provide,
>
> Greg
>
>

```

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