How can I compute the Fourier transform of a unit disk and a unit ball analytically by using Mathematica?

• To: mathgroup at smc.vnet.net
• Subject: [mg55242] How can I compute the Fourier transform of a unit disk and a unit ball analytically by using Mathematica?
• From: Zhou Jiang <jiangzhou_yz at yahoo.com>
• Date: Thu, 17 Mar 2005 03:30:01 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi, My collegues,

I am a fresh bird using Mathematica in my research. I am now trying to compute the analytic expression of the Fourier transform of a unit disk and a unit ball. I hope Mathematica 5.1 can work it out for me just for learning purpose. My definition of the Fourier transform is a little different from the built in function FourierTransfrom. I define the Fourier transform as

F(u)=Integrate[f(x) Exp[-I x u], {x, -Infinity, Infity}]

and the 2D Fourier transform is

F(u,v)=Integrate[f(x,y) Exp[-I (x u+y v)], {x,-Infinity, Infinity},{y,-Infinity, Infinity}]

and hence 3D Fourier transform is defined as

F(u,v,w)=Integrate[f(x,y,z) Exp[-I(x u+y v+z w)], {x,-Infinity, Infinity},{y,-Infinity, Infinity},{z, -Infinity,Infity}]

Therefore, I computed the Fourier transform of the squarewave like

Integrate[Boole[Abs[x]<=1] Exp[- I x u],{x,-Infinity,Infinity}]

and Mathematica 5.1 gave a correct answer  2 sin(u)/u.  I tried the 2D version of the problem as

Integrate[Boole[x^2+y^2<=1] Exp[-I (x u+y v)],{ x,-Infinity, Infity},{y,-Infinity, Infinity}]

and Mathematica cannot work it out any more. I am a little dispressed since this is a very classical problem and the result should be 2 Pi J1[Sqrt[u^2+v^2]]/Sqrt[u^2+v^2] where J1 is the usuall Bessel function of first order. Of course, Mathematica did not give me any answer for the 3D version of the problem. Can anyone give me some idea about how to make it work? Thank you very much.

Daxin

```

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