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Integration test of dirac delta function as a Fourier integral

  1. Sep 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Problem:
    a) Find the Fourier transform of the Dirac delta function: δ(x)
    b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves).
    c) test by integration, that the delta function represented by a Fourier integral integrates to 1

    2. Relevant equations
    So far I've done a) and b) and the delta function turns out to be [itex] \delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega [/itex]

    I've plotted this and it seems to be correct, and I also asked some other students in class and they got the same result, so i don't think that's the issue.

    3. The attempt at a solution
    So to solve c) I try to integrate δ(x) from -∞ to ∞, but it shouldn't really matter as long as 0 is between the integration limits.
    [tex]
    \frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega x}d\omega dx\\
    \frac{1}{2\pi} \int_{-\infty}^{\infty}\left[\frac{1}{ix}e^{i\omega x}\right]_{\infty}^{\infty} dx\\
    \frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\frac{1}{2i}(e^{i\infty x} - e^{-i\infty x}) dx\\
    \frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\sin(\infty x) dx
    [/tex]
    Apparently I end up with an integral that's impossible to solve (without approximation), and the sine function has infinity as its argument... So I was hoping you would know where I went wrong.
     
    Last edited: Sep 19, 2014
  2. jcsd
  3. Sep 19, 2014 #2

    Ray Vickson

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    Science Advisor
    Homework Helper

    The last three lines are nonsense; you need to take a limit. Use
    [tex]\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_N^N e^{i\omega x}d\omega dx[/tex]
    for finite ##N > 0##. Work it through, then take the limit as ##N \to \infty##.

    Hint: Dirichlet Kernel.
     
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