Re: fourier transform of 1/|x|
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Wed, 01 Jul 2009 05:26:15 -0500
On Tue, 30 Jun 2009 22:45:38 -0700 (PDT), norbert9
<norbert9@xxxxxxxxx> wrote:
Indeed, I meant a one-dimensional FT.
Gradshteyn/Ryzhik (4th ed.) list the FT of 1/|x| as 1/|k| in 17.23.
I believe it is correct in some sense.
In what sense? The most general notion of FT I know is the
FT of a tempered distribution, and this is certainly not right
in that sense. (Unless, again, they're talking about R^2.)
Mathematica gives
FourierTransform[1/Abs[x],x,k] = (-2 EulerGamma - 2 Log[Abs[k]])/Sqrt
[2 Pi]
Do you also believe that
1/|k| = (-2 EulerGamma - 2 Log[Abs[k]])/Sqrt[2 Pi]
in some sense?
I'm well aware that \int\cos(kx)/|x| diverges around x=0, but I
suspect there is a way to restrict the support of the function and
define a meaningful FT.
How?
Thanks for any help.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.
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