Example 10. Find the integral representation of the unit function (Heaviside function)

$$
f(t)=\left\{\begin{array}{lll}
0 & \text { for } \quad t<0 \\
1 & \text { for } \quad t>0
\end{array}\right.
$$
Solution. Consider the function

$$
f(t)=\frac{1}{2 \pi i} \int_{C} \frac{e^{-i z t}}{z} d z
$$

where the contour $C$ is shown in Fig. 8.

By closing the contour with a semicircle $C_{R}$ in the upper half-plane, we notice that for $t > 0$, by Jordan's lemma, the integrals

$$
\int_{c_{:}^{\prime}} \frac{e^{-i z t}}{z} d z \rightarrow 0 \quad \text { as } \quad R \rightarrow \infty
$$

Now, the point $z=0$ lies inside the contour of integration. Therefore, by the residue theorem,

$$
f(t)=\frac{1}{2 \pi i} \int_{C} \frac{e^{-i z t}}{z} d z=1 \quad(t>0)
$$

Thus,

$$
f(t)=\frac{1}{2 \pi i} \int_{C} \frac{e^{-i z t}}{z} d z= \begin{cases}0 & \text { for } t < 0 \\ 1 & \text { for } t > 0\end{cases}
$$

Therefore, the considered integral represents a discontinuous function.

## Problems for Independent Solution

Calculate the following integrals:

391. $\int_{-\infty}^{+\infty} \frac{x \cos x d x}{x^{2}-2 x+10}$.
392. $\int_{0}^{\infty} \frac{\cos x d x}{\left(x^{2}+1\right)\left(x^{2}+4\right)}$.
393. $\int_{0}^{\infty} \frac{\cos a x}{1+x^{4}} d x \quad(a>0)$.
394. $\int_{0}^{\infty} \frac{\cos m x}{a^{2}+x^{2}} d x \quad(m>0, a>0)$.
395. $\int_{-\infty}^{+\infty} \frac{\cos \lambda x}{\left(x^{2}+1\right)\left(x^{2}+9\right)} \quad(\lambda>0)$.
396. $\int_{0}^{\infty} \frac{x^{2} \cos x d x}{\left(x^{2}+1\right)^{2}}$.
397. $\int_{-\infty}^{+\infty} \frac{x \sin x}{x^{2}+4 x+20} d x$.
398. $\int_{-\infty}^{+\infty} \frac{\cos x d x}{x^{2}+9}$.
399. $\int_{0}^{\infty} \frac{\cos x}{x^{2}+a^{2}} d x \quad(a>0)$.
400. $\int_{0}^{\infty} \frac{x \sin x}{1+x^{2}+x^{4}} d x$.
401. $\int_{0}^{\infty} \frac{x^{3} \sin a x}{\left(1+x^{2}\right)^{2}} d x \quad(a>0)$.
402. $\int_{0}^{\infty} \frac{3 x^{2}-a^{2}}{\left(x^{2}+b^{2}\right)^{2}} \cos m x d x$.