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Fourier Transformation

  1. Apr 5, 2008 #1
    1. The problem statement, all variables and given/known data
    Generate the fourier transform of the function:
    [tex]f(x)=f_{-Na}(x)+f_{-(N-1)a}(x)+\cdots+f_{(N-1)a}(x)+f_{Na}(x)[/tex]

    2. Relevant equations

    [tex]\tilde{f}(k)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(x)e^{ikx}dx[/tex]
    The following formula may also be used:
    [tex]\frac{1}{x^N}+\frac{1}{x^{N-1}}+\cdots+1+\cdots+x^{N-1}+x^N=x^{-N}(1+x+x^2+x^{2N})
    =\frac{x^{-(N+0.5)}-x^{N+0.5}}{x^{-0.5}-x^{0.5}}[/tex]
    It also says in the question that this is a mathematical form of a diffraction grating.

    3. The attempt at a solution
    My problem with this question is not so much doing the fourier transformation, but more that I don't understant what the subscripts in f(x) mean (for example [tex]f_{-Na}(x)[/tex]), so I can't even make a start on the question without doing it in some general form, so could anybody possibly point me in the right direction with how to tackle these? Any help is greatly appreciated.
     
  2. jcsd
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