Example 1. Find the complex Fourier transform of the function

$$
f(x)=e^{-b^{2} x^{2}}
$$

using the formula

$$
e^{-x^{2}} \longmapsto \frac{1}{\sqrt{2}} e^{-p^{2} / 4}
$$
Solution. Representing the function $f(x)$ as $f(x)=e^{-(b x)^{2}}$, we use property 3. We obtain

$$
e^{-b^{2} x^{2}} \longmapsto \frac{1}{b} \frac{1}{\sqrt{2}} e^{-p^{2} / 4 b^{2}}
$$

Answer. $\quad F(p)=\frac{1}{b \sqrt{2}} e^{-p^{2} /(2 b)^{2}}$.