§1.17 Integral and Series Representations of the Dirac Delta

In applications in physics, engineering, and applied mathematics, (see Friedman (1990)), the Dirac delta distribution (§1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) $\delta \left(x\right)$. This is a symbolic function with the properties:

and

subject to certain conditions on the function $\varphi (x)$. From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that

for a suitably chosen sequence of functions ${\delta }_n\left(x\right)$, $n=1,2,\mathrm{\dots }$. Such a sequence is called a delta sequence and we write, symbolically,

An example of a delta sequence is provided by

In this case

for all functions $\varphi (x)$ that are continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. The last condition is satisfied, for example, when $\varphi (x)=O\left({\mathrm{e}}^{\alpha x^2}\right)$ as $x\to \pm \mathrm{\infty }$, where $\alpha$ is a real constant.

More generally, assume $\varphi (x)$ is piecewise continuous (§1.4(ii)) when $x\in [-c,c]$ for any finite positive real value of $c$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$. Then

Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)):

yields

The inner integral does not converge. However, for $n=1,2,\mathrm{\dots }$,

Hence comparison with (1.17.5) shows that (1.17.9) can be interpreted as a generalized integral (1.17.3) with

provided that $\varphi (x)$ is continuous when $x\in (-\mathrm{\infty },\mathrm{\infty })$, and for each $a$, ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{-n{(x-a)}^2}\varphi (x)dx$ converges absolutely for all sufficiently large values of $n$ (as in the case of (1.17.6)). Then comparison of (1.17.2) and (1.17.9) yields the formal integral representation

Other similar integral representations of the Dirac delta that appear in the physics and applied mathematics literature include the following:

Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)):

yields

The sum ${\sum }_{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}k(x-a)}$ does not converge, but (1.17.18) can be interpreted as a generalized integral in the sense that

where

provided that $\varphi (x)$ is continuous and of period $2\pi$; see §1.8(ii).

By analogy with §1.17(ii) we have the formal ($2\pi$-periodic) series representation

Other similar series representations of the Dirac delta that appear in the physics literature include the following:

The references given in §1.17(ii)–1.17(iii) are from the physics and applied mathematics literature. A comprehensive and detailed applied mathematics approach is that of Friedman (1990, Ch. 3 and 4 ).

For mathematical derivations of the many of the results of §1.17(ii) and §1.17(iii) see Li and Wong (2008). Lebedev (1965) gives an expanded discussion of derivations of (1.17.22)–(1.17.24).