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billet philippe
science forum beginner
Joined: 18 May 2005
Posts: 1
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 Posted: Wed May 18, 2005 7:27 pm Post subject:
Fourier transform
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Hello,
if I choose this formalism for Fourier transform :
Fourier(f(t),x)=integral(f(t)*exp(-i*t*x),t,-infinity,infinity).
What are the Fourier transform of this generalized functions :
1/ log(abs(t)),
2/ abs(t)^alpha,
3/ t^alpha*heaviside(t),
4/ 1/abs(t),
5/ abs(t)^alpha*sign(t).
Thanks for your answers.
Philippe. |
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Paul Abbott
science forum addict
Joined: 19 May 2005
Posts: 99
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| Posted: Thu May 19, 2005 12:48 pm Post subject:
Re: Fourier transform
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In article
<22111558.1116444456582.JavaMail.jakarta@nitrogen.mathforum.org>,
billet philippe <philippe.billet@noos.fr> wrote:
| Quote: | if I choose this formalism for Fourier transform :
Fourier(f(t),x)=integral(f(t)*exp(-i*t*x),t,-infinity,infinity).
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In Mathematica, this definition is specified via
SetOptions[FourierTransform, FourierParameters -> {1, -1}]
| Quote: | What are the Fourier transform of this generalized functions :
1/ log(abs(t)),
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FourierTransform[Log[Abs[t]], t, x]
-2 EulerGamma Pi DiracDelta[x] - Pi/Abs[x]
| Quote: | 3/ t^alpha*heaviside(t),
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For alpha = n a non-negative integer,
I^n (Pi Derivative[n][DiracDelta][x] - (-1)^n I n!/x^(n + 1))
Cheers,
Paul
--
Paul Abbott Phone: +61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul
http://InternationalMathematicaSymposium.org/IMS2005/ |
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Robert B. Israel
science forum Guru
Joined: 24 Mar 2005
Posts: 2151
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| Posted: Thu May 19, 2005 5:30 pm Post subject:
Re: Fourier transform
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In article <paul-47293E.22480519052005@news.uwa.edu.au>,
Paul Abbott <paul@physics.uwa.edu.au> wrote:
| Quote: | In article
22111558.1116444456582.JavaMail.jakarta@nitrogen.mathforum.org>,
billet philippe <philippe.billet@noos.fr> wrote:
if I choose this formalism for Fourier transform :
Fourier(f(t),x)=integral(f(t)*exp(-i*t*x),t,-infinity,infinity).
In Mathematica, this definition is specified via
SetOptions[FourierTransform, FourierParameters -> {1, -1}]
What are the Fourier transform of this generalized functions :
1/ log(abs(t)),
FourierTransform[Log[Abs[t]], t, x]
-2 EulerGamma Pi DiracDelta[x] - Pi/Abs[x]
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Can this be right? It is not a tempered distribution.
| Quote: | 3/ t^alpha*heaviside(t),
For alpha = n a non-negative integer,
I^n (Pi Derivative[n][DiracDelta][x] - (-1)^n I n!/x^(n + 1))
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Similar problem with this when n is odd.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada |
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