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Fourier Transform Help ( f(x) = 1 )

  1. Aug 10, 2011 #1

    FeDeX_LaTeX

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    Hello,

    In the past couple of days I have been looking at how to transform a function f(t) into another function F(s) via the Laplace transform, and have practiced performing simple Laplace transformations such at f(t) = sin(at), sinat, cos(at), eatf(t) and so on.

    I looked on Wikipedia at a formula for the Fourier transform and it's a bit confusing; I know there are several types, here is a formula I found;

    [tex]f(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi ix\xi}dx[/tex] for every real number ξ.

    So if I say that f(x) = 1, then I end up with [tex]f(\xi) = \int_{-\infty}^{\infty}e^{-2\pi ix \xi}dx[/tex].

    Integrating, we get

    [tex]f(\xi) = -\frac{1}{2\pi ix \xi}e^{-2\pi ix \xi}[/tex].

    However I'm stuck on how to evaluate this now, as I have to find the evaluation of the function at negative infinity and subtract that from the function evaluated at infinity (assuming ξ > 0, but when I do that I end up with an [tex]e^{\infty}[/tex] term when evaluating the function at negative infinity. Does this simply mean that it does not converge? Can you show me an example of a function that does converge? I tried using the Fourier transform on sin(x) earlier today and I ended up with [tex]f(\xi) = 0[/tex]...

    Thanks.
     
  2. jcsd
  3. Aug 10, 2011 #2
    The simplest theory I've studied to treat properly Fourier transformation requires that the function you want to transform belongs to L^1 or L^2, where L^1 and L^2 are particular spaces where [itex]f\in L^1 \Leftrightarrow \int_a^b |f|<+\infty[/itex] and [itex]f\in L^2 \Leftrightarrow \int_a^b |f|^2<+\infty[/itex]
    http://en.wikipedia.org/wiki/Lp_space" [Broken]
    http://en.wikipedia.org/wiki/Locally_integrable_function" [Broken]

    For example, you can integrate [itex]e^{-a|x|}\,\,\,a>0[/itex] since it belongs to L^1.
    Other functions that can be integrated are:
    rect(ax)=1 if ax is in [-1/2,1/2], 0 elsewhere
    tri(x)=1-|x|

    To integrate f=1 and f=trigonometric function such sine and cosine, you need a more wide theory, the theory of distribution (I think it is called this way).

    However, if you need, [itex]\mathcal{F}(1)=\delta(x)[/itex].http://en.wikipedia.org/wiki/Dirac_delta_function" [Broken]
     
    Last edited by a moderator: May 5, 2017
  4. Aug 11, 2011 #3

    FeDeX_LaTeX

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    Hello,

    Thanks for the reply. I think I understand what you're trying to say; that the definite integrals from a to b of the functions has to converge (not go to infinity)?

    What are rect(x) and tri(x)?

    I have lectures on Dirac's Delta function and I will look at these soon.

    Thanks for your help.
     
  5. Aug 11, 2011 #4

    hunt_mat

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    I think that rect(x) is the rectangle function, that it s step function that is 1 for a given integral and zero elsewhere, I am guessing that tri(x) is a triangle function which I would imagine is a straight line from (-1,0) to (0,1) and another line from (0,1) to (1,0).

    The Dirac delta function is the answer to your question though.
     
  6. Aug 12, 2011 #5
    I wrote the definitions next to the functions. However, it is as hunt_mat says.
     
  7. Aug 12, 2011 #6

    hunt_mat

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    Wow, excellent guess on my part!!!

    I actually saw the definitions after I posted.
     
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