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

[dimension10]

I would like to know how one finds the Fourier transforms of

$$t$$,

$$\frac{1}{t}$$

and

$${t}^{n}$$

with the definition of the Fourier transform as

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

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

Thanks in advance.

 

[mathman]

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.

 

[Mute]

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

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

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

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

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