% Dirac delta function


# Dirac delta function

The **Dirac delta function** $\delta(x)$, often just called the **delta function**,
is an infinitely narrow discontinuous "spike" at $x = 0$ whose area is
defined to be 1:

$$\begin{aligned}
    \boxed{
        \delta(x) =
        \begin{cases}
            +\infty & \mathrm{if}\: x = 0 \\
            0 & \mathrm{if}\: x \neq 0
        \end{cases}
        \quad \mathrm{and} \quad
        \int_{-\varepsilon}^\varepsilon \delta(x) \dd{x} = 1
    }
\end{aligned}$$

It is sometimes also called the **sampling function**, due to its most
important property: the so-called **sampling property**:

$$\begin{aligned}
    \boxed{
        \int f(x) \: \delta(x - x_0) \: dx = \int f(x) \: \delta(x_0 - x) \: dx = f(x_0)
    }
\end{aligned}$$

$\delta(x)$ is thus an effective weapon against integrals. This may not seem very
useful due to its "unnatural" definition, but in fact it appears as the
limit of several reasonable functions:

$$\begin{aligned}
    \delta(x)
    = \lim_{n \to +\infty} \!\Big\{ \frac{n}{\sqrt{\pi}} \exp(- n^2 x^2) \Big\}
    = \lim_{n \to +\infty} \!\Big\{ \frac{n}{\pi} \frac{1}{1 + n^2 x^2} \Big\}
    = \lim_{n \to +\infty} \!\Big\{ \frac{\sin(n x)}{\pi x} \Big\}
\end{aligned}$$

The last one is especially important, since it is equivalent to the
following integral, which appears very often in the context of
Fourier transforms:

$$\begin{aligned}
    \boxed{
        \delta(x)
        %= \lim_{n \to +\infty} \!\Big\{\frac{\sin(n x)}{\pi x}\Big\}
        = \frac{1}{2\pi} \int_{-\infty}^\infty \exp(i k x) \dd{k}
        \:\:\propto\:\: \hat{\mathcal{F}}\{1\}
    }
\end{aligned}$$

When the argument of $\delta(x)$ is scaled, the delta function is itself scaled:

$$\begin{aligned}
    \boxed{
        \delta(s x) = \frac{1}{|s|} \delta(x)
    }
\end{aligned}$$

*__Proof.__ Because it is symmetric, $\delta(s x) = \delta(|s| x)$. Then by
substituting $\sigma = |s| x$:*

$$\begin{aligned}
    \int \delta(|s| x) \dd{x}
    &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|}
\end{aligned}$$

*__Q.E.D.__*

An even more impressive property is the behaviour of the derivative of
$\delta(x)$:

$$\begin{aligned}
    \boxed{
        \int f(\xi) \: \delta'(x - \xi) \dd{\xi} = f'(x)
    }
\end{aligned}$$

*__Proof.__ Note which variable is used for the
differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:*

$$\begin{aligned}
    \int f(\xi) \: \dv{\delta(x - \xi)}{x} \dd{\xi}
    &= \dv{x} \int f(\xi) \: \delta(x - \xi) \dd{x}
    = f'(x)
\end{aligned}$$

*__Q.E.D.__*

This property also generalizes nicely for the higher-order derivatives:

$$\begin{aligned}
    \boxed{
        \int f(\xi) \: \dv[n]{\delta(x - \xi)}{x} \dd{\xi} = \dv[n]{f(x)}{x}
    }
\end{aligned}$$