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:

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


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