Math Exercises

To understand why the delta function under an integral picks out a particular value of the integrand, it may help to think of it as the limit of a sequence of steps.

Let $\delta(t)$ be the Dirac delta function. We can approximate it with a sequence of rectangular pulses $\delta_n(t)$:

$\delta_n(t) = \begin{cases} h_n, & |t| \le \frac{w_n}{2} \\ 0, & |t| > \frac{w_n}{2} \end{cases}$

Each pulse has area 1: $\int_{-\infty}^{\infty} \delta_n(t) \, dt = h_n w_n = 1$

In the limit, the delta function is obtained as: $\delta(t) = \lim_{n \to \infty} \delta_n(t), \quad w_n \to 0, \quad h_n \to \infty$

The key property under an integral is: $\int_{-\infty}^{\infty} f(t) \, \delta(t-a) \, dt = f(a)$ where $f(t)$ is any continuous function.

Each $\delta_n(t-a)$ is a pulse centered at $t=a$, so in the limit, the integral “picks out” $f(a)$. See a visual below