Dirac delta function

In physics, the Dirac delta function is a function introduced by P.A.M. Dirac in his seminal 1930 book on quantum mechanics. Heuristically, the function can be seen as an extension of the Kronecker delta from discrete to continuous indices. The Kronecker delta acts as a "filter" in a summation:

$\sum _{i=1}^{n}f_{i}\delta _{ia}={\begin{cases}f_{a}&\quad {\hbox{if}}\quad a\in [1,n]\subset \mathbb {N} \\0&\quad {\hbox{if}}\quad a\notin [1,n].\end{cases}}$

Similarly, the Dirac delta function δ(x−a) may be defined by (replace i by x and the summation over i by an integration over x),

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

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. 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 is assumed.

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

The physicist's proof of these properties proceeds by making proper substitutions into the integral and using the ordinary rules of integral calculus.