Example 2. Find the complex Fourier transform of the function

$$
f(x)=\frac{x}{x^{2}+1}
$$

using the formula

$$
\frac{1}{x^{2}+1} \longmapsto \sqrt{\frac{\pi}{2}} e^{-|p|}
$$
Solution. Representing the function $f(x)$ as

$$
f(x)=x \cdot \frac{1}{x^{2}+1}
$$

we use property 6 with $n=1$. We obtain

$$
\frac{x}{x^{2}+1} \longmapsto \frac{1}{i} \frac{d}{d p} \sqrt{\frac{\pi}{2}} e^{-|p|}=-\frac{1}{i} \sqrt{\frac{\pi}{2}} \operatorname{sign}(p) e^{-|p|}
$$

Answer. $F(p)=i \sqrt{\frac{\pi}{2}} \operatorname{sign}(p) e^{-|p|}$.