# §1.17 Integral and Series Representations of the Dirac Delta

## §1.17(i) Delta Sequences

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:

 1.17.1 $\delta\left(x\right)=0,$ $x\in\mathbb{R}$, $x\neq 0$, ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function), $\in$: element of and $\mathbb{R}$: real line Referenced by: §1.17(i) Permalink: http://dlmf.nist.gov/1.17.E1 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

and

 1.17.2 $\int_{-\infty}^{\infty}\delta\left(x-a\right)\phi(x)\,\mathrm{d}x=\phi(a),$ $a\in\mathbb{R}$,

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

 1.17.3 $\lim_{n\to\infty}\int_{-\infty}^{\infty}\delta_{n}\left(x-a\right)\phi(x)\,% \mathrm{d}x=\phi(a),$

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

 1.17.4 $\lim_{n\to\infty}\delta_{n}\left(x\right)=\delta\left(x\right),$ $x\in\mathbb{R}$.

An example of a delta sequence is provided by

 1.17.5 $\delta_{n}\left(x-a\right)=\sqrt{\frac{n}{\pi}}{\mathrm{e}}^{-n(x-a)^{2}}.$ ⓘ Symbols: $\delta_{n}\left(\NVar{x}\right)$: Dirac delta sequence, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: nonnegative integer Referenced by: §1.17(ii) Permalink: http://dlmf.nist.gov/1.17.E5 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

In this case

 1.17.6 $\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x% -a)^{2}}\phi(x)\,\mathrm{d}x=\phi(a),$

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

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

 1.17.7 $\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x% -a)^{2}}\phi(x)\,\mathrm{d}x=\tfrac{1}{2}\phi(a-)+\tfrac{1}{2}\phi(a+).$

## §1.17(ii) Integral Representations

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

 1.17.8 $\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\mathrm{i}at}\left(\int_{-% \infty}^{\infty}\phi(x){\mathrm{e}}^{\mathrm{i}tx}\,\mathrm{d}x\right)\,% \mathrm{d}t=\phi(a)$

yields

 1.17.9 $\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}% ^{\mathrm{i}(x-a)t}\,\mathrm{d}t\right)\phi(x)\,\mathrm{d}x=\phi(a).$

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

 1.17.10 $\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{-t^{2}/(4n)}{\mathrm{e}}^{% \mathrm{i}(x-a)t}\,\mathrm{d}t=\sqrt{\frac{n}{\pi}}{\mathrm{e}}^{-n(x-a)^{2}}.$

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

 1.17.11 $\delta_{n}\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{% -t^{2}/(4n)}{\mathrm{e}}^{\mathrm{i}(x-a)t}\,\mathrm{d}t,$

provided that $\phi(x)$ is continuous when $x\in(-\infty,\infty)$, and for each $a$, $\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x-a)^{2}}\phi(x)\,\mathrm{d}x$ 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

 1.17.12 $\delta\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{% \mathrm{i}(x-a)t}\,\mathrm{d}t.$

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

### Sine and Cosine Functions

 1.17.12_1 $\delta\left(x-a\right)=\frac{2}{\pi}\int_{0}^{\infty}\cos\left(xt\right)\cos% \left(at\right)\,\mathrm{d}t,$ $x>0,a>0$,
 1.17.12_2 $\delta\left(x-a\right)=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(xt\right)\sin% \left(at\right)\,\mathrm{d}t,$ $x>0,a>0$.

Integral representation (1.17.12_1), (1.17.12_2) is the equivalent of the transform pairs, (1.14.9) $\&$ (1.14.11), (1.14.10) $\&$ (1.14.12), respectively. See Friedman (1990, p. 250).

### Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))

 1.17.13 $\delta\left(x-a\right)=x\int_{0}^{\infty}tJ_{\nu}\left(xt\right)J_{\nu}\left(% at\right)\,\mathrm{d}t,$ $\Re\nu>-1$, $x>0$, $a>0$,
 1.17.14 $\delta\left(x-a\right)=\frac{2xa}{\pi}\int_{0}^{\infty}t^{2}\mathsf{j}_{\ell}% \left(xt\right)\mathsf{j}_{\ell}\left(at\right)\,\mathrm{d}t,$ $x>0$, $a>0$.

See Arfken and Weber (2005, Eq. (11.59)) and Konopinski (1981, p. 242). For a generalization of (1.17.14) see Maximon (1991).

### Coulomb Functions (§33.14(iv))

 1.17.15 $\delta\left(x-a\right)=\int_{0}^{\infty}s\left(x,\ell;r\right)s\left(a,\ell;r% \right)\,\mathrm{d}r,$ $a>0$, $x>0$.

See Seaton (2002a).

### Airy Functions (§9.2)

 1.17.16 $\delta\left(x-a\right)=\int_{-\infty}^{\infty}\operatorname{Ai}\left(t-x\right% )\operatorname{Ai}\left(t-a\right)\,\mathrm{d}t.$

See Vallée and Soares (2010, §3.5.3).

In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^{2}$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv).

## §1.17(iii) Series Representations

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

 1.17.17 $\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}{\mathrm{e}}^{-\mathrm{i}ka}\left(\int_% {-\pi}^{\pi}\phi(x){\mathrm{e}}^{\mathrm{i}kx}\,\mathrm{d}x\right)=\phi(a),$

yields

 1.17.18 $\int_{-\pi}^{\pi}\phi(x)\left(\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}{\mathrm{% e}}^{\mathrm{i}k(x-a)}\right)\,\mathrm{d}x=\phi(a).$

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

 1.17.19 $\lim_{n\to\infty}\int_{-\pi}^{\pi}\delta_{n}\left(x-a\right)\phi(x)\,\mathrm{d% }x=\phi(a),$

where

 1.17.20 $\delta_{n}\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-n}^{n}{\mathrm{e}}^{\mathrm{% i}k(x-a)}\left(=\frac{\sin\left((n+\frac{1}{2})(x-a)\right)}{2\pi\sin\left(% \frac{1}{2}(x-a)\right)}\right),$

provided that $\phi(x)$ 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

 1.17.21 $\delta\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}{\mathrm{e}}^{% \mathrm{i}k(x-a)}.$

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

### Legendre Polynomials (§§14.7(i) and 18.3)

 1.17.22 $\delta\left(x-a\right)=\sum_{k=0}^{\infty}(k+\tfrac{1}{2})P_{k}\left(x\right)P% _{k}\left(a\right).$ ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function), $P_{\NVar{n}}\left(\NVar{x}\right)$: Legendre polynomial and $k$: integer Referenced by: §1.17(iii), §1.17(iv), §14.18(iii) Permalink: http://dlmf.nist.gov/1.17.E22 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

### Laguerre Polynomials (§18.3)

 1.17.23 $\delta\left(x-a\right)={\mathrm{e}}^{-(x+a)/2}\sum_{k=0}^{\infty}L_{k}\left(x% \right)L_{k}\left(a\right).$

### Hermite Polynomials (§18.3)

 1.17.24 $\delta\left(x-a\right)=\frac{{\mathrm{e}}^{-(x^{2}+a^{2})/2}}{\sqrt{\pi}}\sum_% {k=0}^{\infty}\frac{H_{k}\left(x\right)H_{k}\left(a\right)}{2^{k}k!}.$

### Spherical Harmonics (§14.30)

 1.17.25 $\delta\left(\cos\theta_{1}-\cos\theta_{2}\right)\delta\left(\phi_{1}-\phi_{2}% \right)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}Y_{{\ell},{m}}\left(\theta_% {1},\phi_{1}\right)\overline{Y_{{\ell},{m}}\left(\theta_{2},\phi_{2}\right)}.$ ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$: Dirac delta (or Dirac delta function), $\overline{\NVar{z}}$: complex conjugate, $\cos\NVar{z}$: cosine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic and $m$: nonnegative integer Referenced by: §1.17(iii) Permalink: http://dlmf.nist.gov/1.17.E25 Encodings: TeX, pMML, png Notational Change (effective with 1.0.19): As a notational clarification, the second spherical harmonic in the sum over $\ell$ has been explicity marked up as a complex conjugate. See also: Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

(1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). For (1.17.25) see Arfken and Weber (2005, p. 792).

## §1.17(iv) Mathematical Definitions

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).