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

  • #1
371
0

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
mathman
Science Advisor
7,760
415
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.
 
  • #3
Mute
Homework Helper
1,388
10
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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