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Mistake in book?

  1. Dec 12, 2011 #1
    In book Modern theory of critical phenomena author Shang - Keng Ma in page 17.

    [tex]\sigma_{\vec{k}}=V^{-\frac{1}{2}}\int d^3\vec{x}e^{-i\vec{k}\cdot\vec{x}}\sigma(\vec{x})[/tex]

    [tex]\sigma(\vec{x})=V^{-\frac{1}{2}}\sum_{\vec{k}}e^{i\vec{k}\cdot \vec{x}}\sigma_{\vec{k}}[/tex]

    Is this correct? How can inversion of continual FT be discrete FT? Thanks for your answer.
  2. jcsd
  3. Dec 12, 2011 #2


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    The sum is to be understood as an integral over a comb of very narrow functions which approach delta functions in the limit V to infinity, i.e. think of the FT of a product of a periodic function with a rectangle of width V.
    The integral over the delta functions is then equivalent to a sum over their locations.
  4. Dec 12, 2011 #3
    If the x-domain is bounded, the k-domain will be discrete. This is what happens for an ordinary Fourier series for functions on a bounded domain, or periodic:

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