Categories: Mathematics, Physics.

# Dirac delta function

The **Dirac delta function** $\delta(x)$, often just the **delta function**,
is a function (or, more accurately, a Schwartz distribution)
that is commonly used in physics.
It is an infinitely narrow discontinuous “spike” at $x = 0$ whose area is
defined to be 1:

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

$\delta(x)$ is thus quite 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:

$\begin{aligned} \delta(x) = \lim_{n \to +\infty} \!\Big\{\frac{\sin(n x)}{\pi x}\Big\} = \frac{1}{2\pi} \int_{-\infty}^\infty \exp(i k x) \dd{k} \:\:\propto\:\: \hat{\mathcal{F}}\{1\} \end{aligned}$When the argument of $\delta(x)$ is scaled, the delta function is itself scaled:

$\begin{aligned} \boxed{ \delta(s x) = \frac{1}{|s|} \delta(x) } \end{aligned}$Because it is symmetric, $\delta(s x) = \delta(|s| x)$. Then by substituting $\sigma = |s| x$:

$\begin{aligned} \int \delta(|s| x) \dd{x} &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|} \end{aligned}$An even more impressive property is the behaviour of the derivative of $\delta(x)$:

$\begin{aligned} \boxed{ \int f(\xi) \: \delta'(x - \xi) \dd{\xi} = f'(x) } \end{aligned}$Note which variable is used for the differentiation, and that $\delta'(x - \xi) = - \delta'(\xi - x)$:

$\begin{aligned} \int f(\xi) \: \dv{\delta(x - \xi)}{x} \dd{\xi} &= \dv{}{x}\int f(\xi) \: \delta(x - \xi) \dd{x} = f'(x) \end{aligned}$This property also generalizes nicely for the higher-order derivatives:

$\begin{aligned} \boxed{ \int f(\xi) \: \dvn{n}{\delta(x - \xi)}{x} \dd{\xi} = \dvn{n}{f(x)}{x} } \end{aligned}$## References

- O. Bang,
*Applied mathematics for physicists: lecture notes*, 2019, unpublished.