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.