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Fourier transform of 1

  1. Nov 22, 2011 #1
    How does one find the Fourier Transform of 1?

    [tex]\mathscr{F}\{1\}=\mathcal{F}\{1\}=\int\limits_{-\infty}^{\infty}{e}^{-i \omega t} \mbox{d}t=?[/tex]

    I tried to solve it and came up with

    [tex]\sqrt{\frac{2}{\pi}}\frac{1}{\omega}\lim_{t \rightarrow \infty}\sin\left(\omega t\right) [/tex]

    but that is indeterminate whereas actual answer is

    [tex]\sqrt{2\pi}\delta\left(\omega\right)[/tex]

    So how does one actually solve this Fourier Transform.

    Thanks in advance.
     
    Last edited: Nov 22, 2011
  2. jcsd
  3. Nov 22, 2011 #2
    Use duality. Compute the Fourier transform of the delta function and you get a constant, so the Fourier transform of a constant is a delta function. Just look up duality for Fourier transforms and you'll see what I mean.
     
  4. Nov 22, 2011 #3
    Thanks a lot!
     
    Last edited: Nov 22, 2011
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