Example 3. Find the complex Fourier transform of the function

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

using the formula

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

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

we use property 4 with $c=1$. We obtain

$$
\frac{1}{x^{2}+2 x+2} \longmapsto e^{-i p} \sqrt{\frac{\pi}{2}} e^{-|p|}
$$

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

4.4. Complex transformation of the function $\sum_{k-1}^{n} a_{k} x^{m_{k}} f_{k}\left(b_{k} x+c_{k}\right) \quad 145$