Fourier transform of 1

  • #1
371
0

Main Question or Discussion Point

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:

Answers and Replies

  • #2
1,761
123
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.
 
  • #3
371
0
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.
Thanks a lot!
 
Last edited:

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