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The Dirac delta function is a function introduced in 1930 by Paul Adrien Maurice Dirac in his seminal book on quantum mechanics. A physical model that visualizes a delta function is a mass distribution of finite total mass M—the integral over the mass distribution. When the distribution becomes smaller and smaller, while M is constant, the mass distribution shrinks to a point mass, which by definition has zero extent and yet has a finite-valued integral equal to total mass M. In the limit of a point mass the distribution becomes a Dirac delta function.

Heuristically, the Dirac delta function can be seen as an extension of the Kronecker delta from integral indices (elements of $\mathbb {Z}$) to real indices (elements of $\mathbb {R}$). Note that the Kronecker delta acts as a "filter" in a summation:

$\sum _{i=m}^{n}\;f_{i}\;\delta _{ia}={\begin{cases}f_{a}&\quad {\hbox{if}}\quad a\in [m,n]\subset \mathbb {Z} \\0&\quad {\hbox{if}}\quad a\notin [m,n].\end{cases}}$

In analogy, the Dirac delta function δ(x−a) is defined by (replace i by x and the summation over i by an integration over x),

$\int _{a_{0}}^{a_{1}}f(x)\delta (x-a)\mathrm {d} x={\begin{cases}f(a)&\quad {\hbox{if}}\quad a\in [a_{0},a_{1}]\subset \mathbb {R} ,\\0&\quad {\hbox{if}}\quad a\notin [a_{0},a_{1}].\end{cases}}$

The Dirac delta function is not an ordinary well-behaved map $\mathbb {R} \rightarrow \mathbb {R}$, but a distribution, also known as an improper or generalized function. Physicists express its special character by stating that the Dirac delta function makes only sense as a factor in an integrand ("under the integral"). Mathematicians say that the delta function is a linear functional on a space of test functions.

Properties

Most commonly one takes the lower and the upper bound in the definition of the delta function equal to $-\infty$ and $\infty$, respectively. From here on this will be done.

${\begin{aligned}\int _{-\infty }^{\infty }\delta (x)\mathrm {d} x&=1,\\{\frac {1}{2\pi }}\int _{-\infty }^{\infty }e^{ikx}\mathrm {d} k&=\delta (x)\\\delta (x-a)&=\delta (a-x),\\(x-a)\delta (x-a)&=0,\\\delta (ax)&=|a|^{-1}\delta (x)\quad (a\neq 0),\\f(x)\delta (x-a)&=f(a)\delta (x-a),\\\int _{-\infty }^{\infty }\delta (x-y)\delta (y-a)\mathrm {d} y&=\delta (x-a)\end{aligned}}$

The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus. The delta function as a Fourier transform of the unit function f(x) = 1 (the second property) will be proved below. The last property is the analogy of the multiplication of two identity matrices,

$\sum _{j=1}^{n}\;\delta _{ij}\;\delta _{jk}=\delta _{ik},\quad i,k=1,\ldots ,n.$

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