Fourier transform of f(x)/x

Question

Is there a relation to the Fourier transform of the following statement? $$\tau(\frac{f(x)}{x})=??$$ and $$\tau(\frac{f(x)}{x^{2}})=??$$ Is this correct? $$\tau(\frac{f(x)}{x})=\tau(f(x))\tau(\frac{1}{x})$$ I want to know if there is a relationship like this? $$\tau(x^{n} f(x))=(-i)^{n}\frac{d^{n}}{dk^{n}}F(k)$$

Answer 1

Let $\mathcal F_{x\to\xi}\{f(x)\} = \int_{-\infty}^{\infty} f(x) \, e^{-i\xi x} \, dx.$ Then we have $\mathcal F_{x\to\xi}\{x \, f(x)\} = i \frac{d}{d\xi} \mathcal F_{x\to\xi}\{f(x)\}.$ This implies that $$\mathcal F_{x\to\xi}\{f(x)\} = \mathcal F_{x\to\xi}\left\{ x \, \frac{f(x)}{x} \right\} = i \frac{d}{d\xi} \mathcal F_{x\to\xi}\left\{ \frac{f(x)}{x} \right\}$$

Therefore $$ \mathcal F_{x\to\xi}\left\{ \frac{f(x)}{x} \right\} = -i \int \mathcal F_{x\to\xi}\{f(x)\} \, d\xi + C.$$