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Definition clarification for Fourier transform

  1. Sep 20, 2015 #1
    I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem:

    In some sources, I see the forward and inverse transforms defined as such:
    F(k) = [1/(√2π)] ∫-∞ f(x)eikx dx
    f(x) = [1/(√2π)] ∫-∞ f(k)eikx dk

    In other cases, I've seen:
    F(k) = [1/(√2π)] ∫-∞ f(x)e-ikx dx
    f(x) = [1/(√2π)] ∫-∞ f(k)eikx dk

    Notice that in the first version of the forward transform (the one that solves for F(k)), the exponential in the integrand has a positive sign in the exponent ikx, while in the 2nd version it has a negative ikx.

    Which version is correct? Are they both correct and it is a matter of convention? Are neither correct?

    Also, is there some way to do a multiple dimensional Fourier transform using volume integrals? If so, what is the formula for that (preferably including (√2π))?
  2. jcsd
  3. Sep 20, 2015 #2
    Only the 2nd pair is correct. There are a couple conventional issues, but no matter what the sign on the exponent has to change for the inverse transform relative to the forward transform.
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