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Fourier transform of t, 1/t and t^n

  1. Nov 24, 2011 #1
    I would like to know how one finds the Fourier transforms of

    [tex]t[/tex],

    [tex]\frac{1}{t}[/tex]

    and

    [tex]{t}^{n}[/tex]

    with the definition of the fourier transform as

    [tex]\mathscr{F}\{f(t)\}=\mathcal{F}\{f(t)\}=\frac{1}{ \sqrt{2\pi} }\int\limits_{-\infty}^{\infty}{e}^{-i\omega t}f(t)\mbox{d}t[/tex]

    I have tried the definition of a fourier transform and I got some weird limits. Laplace transforms are so much easier!

    Thanks in advance.
     
  2. jcsd
  3. Nov 24, 2011 #2

    mathman

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    Science Advisor
    Gold Member

    In the usual definition of Fourier transform, f(t) is usually presumed to be integrable, or square integrable. None of your functions satisfy this requirement.
     
  4. Nov 25, 2011 #3

    Mute

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    Homework Helper

    The functions do, however, have Fourier transforms in terms of distributions. Consider

    [tex]\sqrt{2\pi}\delta(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty dt~e^{i\omega t}.[/tex]

    Now, take a derivative of both sides with respect to the frequency:

    [tex]\sqrt{2\pi}\delta'(\omega) = \frac{i}{\sqrt{2\pi}} \int_{-\infty}^\infty dt~t e^{i\omega t}.[/tex]

    You can keep taking derivatives to get the Fourier transform of tn. For 1/t, the fourier transform will be proportional to the [itex]\mbox{sgn}(\omega)[/itex] function, where sgn(x) returns the sign of x.
     
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