Example 3. Find the function $f(x)$, the complex Fourier transform of which is

$$
F(p)=\left[\begin{array}{ll}
1, & 0<p<1 \\
0, & 1<p<\infty
\end{array}\right.
$$
SOLUTION.

1. We check that condition (4) is satisfied:

$$
\int_{-\infty}^{+\infty}|F(p)|^{2} d p=\int_{0}^{1} d p=1<\infty
$$

2. By computing the integral on the right-hand side of (5), we obtain:

$$
f(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+\infty} F(p) e^{-i p x} d p=\frac{1}{\sqrt{2 \pi}} \int_{0}^{1} e^{-i p x} d p=\frac{1}{\sqrt{2 \pi}} \frac{1-e^{-i x}}{i x}
$$

Answer. $f(x)=\frac{1}{\sqrt{2 \pi}} \frac{1-e^{-i x}}{i x}$.