[Math] Fourier Transform of $|x|^n$

dirac deltafourier analysis

I know that Fourier transform of $x^n \Leftrightarrow2\pi i^n \delta^{(n)}(\omega)$ where $n$ is an integer.
I was wondering how is this effect if we want to find Fourier of $|x|^n$? Of course, if $n$ is even nothing changes, what if $n$ is odd.
I looked at table of transforms, but was not able to find it.

Best Answer

Hint:

Define a function

$$f(x)=x^nu(x)$$

where $u(x)$ is the unit step function, and note that

$$|x|^n=f(x)+f(-x)\tag{1}$$

From (1), you get the Fourier transform pair

$$|x|^n\Longleftrightarrow F(\omega)+F(-\omega)=2\text{Re}\{F(\omega)\}\tag{2}$$

You can compute $F(\omega)$ from

$$F(\omega)=\mathcal{F}\{x^nu(x)\}=\frac{1}{2\pi}\mathcal{F}\{x^n\}*\mathcal{F}\{u(x)\}\tag{3}$$

where $*$ denotes convolution. The only thing you need to evaluate (3) is $\mathcal{F}\{u(x)\}$, which you can find in almost all Fourier transform tables.

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