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Robinett's fourier transform

  1. Sep 16, 2007 #1
    1. The problem statement, all variables and given/known data
    Richard Robinett defined the Fourier transform with an exp(-ikx) and the inverse Fourier transform with an exp(ikx). I have always seen the opposite convention and I thought it was not even a convention but a necessity to do it the other in order to apply it to some Gaussian equations. Has anyone ever seen this sign convention before?




    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 16, 2007 #2

    dextercioby

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    It really isn't relevant whether it's with a plus, or with a minus. I've seen in most cases

    [tex]\phi (x)=\frac{1}{(2\pi)^{3/2}}\int dk \ \tilde{\phi}(k) e^{-ikx} [/tex].
     
  4. Sep 16, 2007 #3

    HallsofIvy

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    Note that the integration is from [itex]-\infty[/itex] to [itex]\infty[/itex]. That's why the sign does not matter.
     
  5. Sep 16, 2007 #4
    Furthermore, the constants in front also do not really matter, as long as they combine to give 1/(2 pi). There are a couple of theorems which depend on them (I think Parseval's theorem and the associated ones do), but it's all up to a constant. My supervisor (in physics) recommends just ignoring the constants, and adding them back in if you have to at the end :wink:
     
  6. Sep 16, 2007 #5
    If you define the Fourier transform as dextercioby did:
    [tex]\phi (x)=\frac{1}{(2\pi)^{3/2}}\int dk \phi(k) e^{-ikx}[/tex]
    then the inverse transform is:
    [tex]\phi (k)=\frac{1}{(2\pi)^{3/2}}\int dx \phi(x) e^{ikx}[/tex]
    It is merely a matter of convention which is called which. There's no 'wrong' convention as long as you remain consistent.

    On page 11 of 'Photons and Atoms' by Claude Cohen-Tannoudji et. al. the convention above is used. On page 97 of 'The Principles of QM' by P.A.M. Dirac, the convention is left deliciously ambiguous.
     
    Last edited: Sep 16, 2007
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