Fourier transform of mod of a function

Question

How to calculate the Fourier transform of $$f(x)=\frac{1}{|x+c||x-c|}$$ I tried using " https://www.wolframalpha.com/input/?i=Fourier+transform+calculator", but I don't get any results, please help me.

Answer 1

Almost an answer, but with remaining questions...

First of all, this issue has to be considered in the sense of distributions.

Let us first concentrate on the F.T. of $1/|x|$ which is (see formula (2) in the detailed answer of @Sangchul Lee to this question:

$$\mathcal F (\tfrac{1}{|x|})=- \log|\xi| - (\gamma + \log(2\pi))\delta \tag{1}$$

where $\gamma$ is the Euler-Mascheroni constant.

Out of (1), it is immediate to obtain $\mathcal F (1/|x-c|)$ and $\mathcal F (1/|x+c|)$; it remains to convolve them, because a product is transformed into a convolution by $\mathcal F$.

Now, can we obtain a final closed form ? Maybe by working on the result obtained here ?

Using DFT, I have obtained the following ogival curve for this Fourier transform (code below); I will continue to work on this question ...

Matlab code:

x=-20:0.011:20;
c=1; % other values give almost the same shape
s=1./abs(x-c);u=fftshift(abs(fft(s)));
s=1./abs(x+c);v=fftshift(abs(fft(s)));
w=conv(u,v,'same');
plot(x,1e-7*w);

Remark: Wolfram Alpha, to the question phrased in this way: FourierTransform[1/Abs[x]], does provide an answer:

$$\tfrac{1}{\sqrt{2 \pi}} \int \tfrac{1}{|x|} e^{i\omega x}dx=-\sqrt{\tfrac{2}{\pi}}\left(\log(|\omega|)+\gamma\right)$$

The normalization factor $\tfrac{1}{\sqrt{2 \pi}}$ isn't an issue, we know it (it is one of the "avatars" of the Fourier Transform); but where have the $2 \pi$ and the dirac $\delta$ gone? ...