1 Algebraic and Analytic Methods Topics of Discussion 1.16 Distributions 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.17 Integral and Series Representations of the Dirac Delta

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Keywords:: Dirac delta, Dirac delta function
Referenced by:: §1.18(ii), §1.4(v), Ch.1, §18.1(i), §20.13, §33.1, §33.14(iv), Erratum (V1.2.0) §1.17, Methodology
Permalink:: http://dlmf.nist.gov/1.17
See also:: Annotations for Ch.1

§1.17(i) Delta Sequences
§1.17(ii) Integral Representations
§1.17(iii) Series Representations
§1.17(iv) Mathematical Definitions

§1.17(i) Delta Sequences

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Defines:: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function) and $\delta_{n}\left(\NVar{x}\right)$ : Dirac delta sequence
Keywords:: Dirac delta, delta sequence, delta sequences
Notes:: (1.17.6) is a special case of Theorem 7.1 of Olver (1997b, Chapter 3) when $\phi(a)\neq 0$ . This theorem also extends straightforwardly to cover $\phi(a)=0$ . (1.17.7) is proved in a similar manner.
Referenced by:: §1.16(iii), §1.16(iii)
Permalink:: http://dlmf.nist.gov/1.17.i
Addition (effective with 1.2.0):: In the first sentence, a reference to Friedman (1990) was added.
Modification (effective with 1.2.0):: Immediately above (1.17.1) “operator” was replaced with “symbolic function”.
See also:: Annotations for §1.17 and Ch.1

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		$\int_{-\infty}^{\infty}\delta\left(x-a\right)\phi(x)\,\mathrm{d}x=\phi(a),$
$a\in\mathbb{R}$ ,
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\mathbb{R}$ : real line and $\phi(x)$ : continuous function Referenced by: §1.16(viii), §1.17(i), §1.17(ii) Permalink: http://dlmf.nist.gov/1.17.E2 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

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		$\lim_{n\to\infty}\int_{-\infty}^{\infty}\delta_{n}\left(x-a\right)\phi(x)\,% \mathrm{d}x=\phi(a),$
ⓘ Symbols: $\delta_{n}\left(\NVar{x}\right)$ : Dirac delta sequence, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\phi(x)$ : continuous function Referenced by: §1.17(ii) Permalink: http://dlmf.nist.gov/1.17.E3 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

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		$\lim_{n\to\infty}\delta_{n}\left(x\right)=\delta\left(x\right),$
$x\in\mathbb{R}$ .
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\delta_{n}\left(\NVar{x}\right)$ : Dirac delta sequence, $\in$ : element of, $\mathbb{R}$ : real line and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.17.E4 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

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		$\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x% -a)^{2}}\phi(x)\,\mathrm{d}x=\phi(a),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $n$ : nonnegative integer and $\phi(x)$ : continuous function Referenced by: §1.17(i), §1.17(ii), §7.19(iii) Permalink: http://dlmf.nist.gov/1.17.E6 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

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		$\lim_{n\to\infty}\sqrt{\frac{n}{\pi}}\int_{-\infty}^{\infty}{\mathrm{e}}^{-n(x% -a)^{2}}\phi(x)\,\mathrm{d}x=\tfrac{1}{2}\phi(a-)+\tfrac{1}{2}\phi(a+).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $n$ : nonnegative integer and $\phi(x)$ : continuous function Referenced by: §1.17(i) Permalink: http://dlmf.nist.gov/1.17.E7 Encodings: TeX, pMML, png See also: Annotations for §1.17(i), §1.17 and Ch.1

§1.17(ii) Integral Representations

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Keywords:: Dirac delta, Fourier, Fourier integral, Fourier series, integral representations
Notes:: For (1.17.10) complete the square in the total power of $\mathrm{e}$ , make the change of variable $\tau=(t/(2\sqrt{n}))-\mathrm{i}(x-a)\sqrt{n}$ , and use $\int_{-\infty}^{\infty}{\mathrm{e}}^{-\tau^{2}}\,\mathrm{d}\tau=\sqrt{\pi}$ .
Referenced by:: §1.17(iii), §1.17(iv), §1.17(iv), §10.22(iv), §10.59, §9.11(iii), Erratum (V1.2.0) §1.17
Permalink:: http://dlmf.nist.gov/1.17.ii
See also:: Annotations for §1.17 and Ch.1

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

1.17.8		$\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{-\mathrm{i}at}\left(\int_{-% \infty}^{\infty}\phi(x){\mathrm{e}}^{\mathrm{i}tx}\,\mathrm{d}x\right)\,% \mathrm{d}t=\phi(a)$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\phi(x)$ : continuous function Referenced by: §1.16(viii) Permalink: http://dlmf.nist.gov/1.17.E8 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17 and Ch.1

yields

1.17.9		$\int_{-\infty}^{\infty}\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}% ^{\mathrm{i}(x-a)t}\,\mathrm{d}t\right)\phi(x)\,\mathrm{d}x=\phi(a).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\phi(x)$ : continuous function Referenced by: §1.17(ii) Permalink: http://dlmf.nist.gov/1.17.E9 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17 and Ch.1

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

1.17.10		$\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{-t^{2}/(4n)}{\mathrm{e}}^{% \mathrm{i}(x-a)t}\,\mathrm{d}t=\sqrt{\frac{n}{\pi}}{\mathrm{e}}^{-n(x-a)^{2}}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $n$ : nonnegative integer Referenced by: §1.17(ii) Permalink: http://dlmf.nist.gov/1.17.E10 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17 and Ch.1

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		$\delta_{n}\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{% -t^{2}/(4n)}{\mathrm{e}}^{\mathrm{i}(x-a)t}\,\mathrm{d}t,$
ⓘ Symbols: $\delta_{n}\left(\NVar{x}\right)$ : Dirac delta sequence, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.17.E11 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17 and Ch.1

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		$\delta\left(x-a\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}{\mathrm{e}}^{% \mathrm{i}(x-a)t}\,\mathrm{d}t.$
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $\int$ : integral Referenced by: §1.16(viii) Permalink: http://dlmf.nist.gov/1.17.E12 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17 and Ch.1

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

Sine and Cosine Functions

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Keywords:: Cosines, Dirac delta, Sines, integral representations
Addition (effective with 1.2.0):: This paragraph including (1.17.12_1), (1.17.12_2) were added.
See also:: Annotations for §1.17(ii), §1.17 and Ch.1

1.17.12_1		$\delta\left(x-a\right)=\frac{2}{\pi}\int_{0}^{\infty}\cos\left(xt\right)\cos% \left(at\right)\,\mathrm{d}t,$
$x>0,a>0$ ,
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Referenced by: §1.17(ii), §1.17(ii), §1.17(ii), §1.18(vi), Erratum (V1.2.0) §1.17 Permalink: http://dlmf.nist.gov/1.17.E12_1 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

1.17.12_2		$\delta\left(x-a\right)=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(xt\right)\sin% \left(at\right)\,\mathrm{d}t,$
$x>0,a>0$ .
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function Referenced by: §1.17(ii), §1.17(ii), §1.17(ii), §1.18(vi), Erratum (V1.2.0) §1.17 Permalink: http://dlmf.nist.gov/1.17.E12_2 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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))

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Keywords:: Bessel functions, Dirac delta, integral representations, spherical Bessel functions
See also:: Annotations for §1.17(ii), §1.17 and Ch.1

1.17.13		$\delta\left(x-a\right)=x\int_{0}^{\infty}tJ_{\nu}\left(xt\right)J_{\nu}\left(% at\right)\,\mathrm{d}t,$
$\Re\nu>-1$ , $x>0$ , $a>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\Re$ : real part Referenced by: §1.18(vi) Permalink: http://dlmf.nist.gov/1.17.E13 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

1.17.14		$\delta\left(x-a\right)=\frac{2xa}{\pi}\int_{0}^{\infty}t^{2}\mathsf{j}_{\ell}% \left(xt\right)\mathsf{j}_{\ell}\left(at\right)\,\mathrm{d}t,$
$x>0$ , $a>0$ .
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind Referenced by: §1.17(ii), §1.18(vi) Permalink: http://dlmf.nist.gov/1.17.E14 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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))

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Keywords:: Coulomb functions, Dirac delta, integral representations
See also:: Annotations for §1.17(ii), §1.17 and Ch.1

1.17.15		$\delta\left(x-a\right)=\int_{0}^{\infty}s\left(x,\ell;r\right)s\left(a,\ell;r% \right)\,\mathrm{d}r,$
$a>0$ , $x>0$ .
ⓘ Symbols: $s\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Permalink: http://dlmf.nist.gov/1.17.E15 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

See Seaton (2002a).

Airy Functions (§9.2)

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Keywords:: Airy functions, Dirac delta, integral representations
Addition (effective with 1.2.0):: Three sentences were added at the end of this paragraph.
See also:: Annotations for §1.17(ii), §1.17 and Ch.1

1.17.16		$\delta\left(x-a\right)=\int_{-\infty}^{\infty}\operatorname{Ai}\left(t-x\right% )\operatorname{Ai}\left(t-a\right)\,\mathrm{d}t.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\,\mathrm{d}\NVar{x}$ : differential of $x$ and $\int$ : integral Referenced by: §1.17(ii) Permalink: http://dlmf.nist.gov/1.17.E16 Encodings: TeX, pMML, png See also: Annotations for §1.17(ii), §1.17(ii), §1.17 and Ch.1

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

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Keywords:: Dirac delta, Fourier, series representations
Referenced by:: §1.17(iv), §1.17(iv), §14.30(iii), Erratum (V1.0.19) for Notation, Erratum (V1.2.1) for §1.17(iii)
Permalink:: http://dlmf.nist.gov/1.17.iii
Clarification (effective with 1.2.1):: Just above (1.17.21), “formal” has been replaced with “formal ( $2\pi$ -periodic)”.
Suggested 2023-08-23 by Scott Glancy
See also:: Annotations for §1.17 and Ch.1

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

1.17.17		$\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}{\mathrm{e}}^{-\mathrm{i}ka}\left(\int_% {-\pi}^{\pi}\phi(x){\mathrm{e}}^{\mathrm{i}kx}\,\mathrm{d}x\right)=\phi(a),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $k$ : integer and $\phi(x)$ : continuous function Permalink: http://dlmf.nist.gov/1.17.E17 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17 and Ch.1

yields

1.17.18		$\int_{-\pi}^{\pi}\phi(x)\left(\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}{\mathrm{% e}}^{\mathrm{i}k(x-a)}\right)\,\mathrm{d}x=\phi(a).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $k$ : integer and $\phi(x)$ : continuous function Referenced by: §1.17(iii) Permalink: http://dlmf.nist.gov/1.17.E18 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17 and Ch.1

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		$\lim_{n\to\infty}\int_{-\pi}^{\pi}\delta_{n}\left(x-a\right)\phi(x)\,\mathrm{d% }x=\phi(a),$
ⓘ Symbols: $\delta_{n}\left(\NVar{x}\right)$ : Dirac delta sequence, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $n$ : nonnegative integer and $\phi(x)$ : continuous function Permalink: http://dlmf.nist.gov/1.17.E19 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17 and Ch.1

where

1.17.20		$\delta_{n}\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-n}^{n}{\mathrm{e}}^{\mathrm{% i}k(x-a)}\left(=\frac{\sin\left((n+\frac{1}{2})(x-a)\right)}{2\pi\sin\left(% \frac{1}{2}(x-a)\right)}\right),$
ⓘ 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, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, $k$ : integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/1.17.E20 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17 and Ch.1

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		$\delta\left(x-a\right)=\frac{1}{2\pi}\sum_{k=-\infty}^{\infty}{\mathrm{e}}^{% \mathrm{i}k(x-a)}.$
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $k$ : integer Referenced by: §1.17(iii), §1.18(v), Erratum (V1.2.1) for §1.17(iii) Permalink: http://dlmf.nist.gov/1.17.E21 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17 and Ch.1

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)

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Keywords:: Dirac delta, Legendre polynomials, series representations
See also:: Annotations for §1.17(iii), §1.17 and Ch.1

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)

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Keywords:: Dirac delta, Laguerre polynomials, series representations
See also:: Annotations for §1.17(iii), §1.17 and Ch.1

1.17.23		$\delta\left(x-a\right)={\mathrm{e}}^{-(x+a)/2}\sum_{k=0}^{\infty}L_{k}\left(x% \right)L_{k}\left(a\right).$
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $\mathrm{e}$ : base of natural logarithm, $L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)$ : Laguerre polynomial and $k$ : integer Permalink: http://dlmf.nist.gov/1.17.E23 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

Hermite Polynomials (§18.3)

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Keywords:: Dirac delta, Hermite polynomials, series representations
See also:: Annotations for §1.17(iii), §1.17 and Ch.1

1.17.24		$\delta\left(x-a\right)=\frac{{\mathrm{e}}^{-(x^{2}+a^{2})/2}}{\sqrt{\pi}}\sum_% {k=0}^{\infty}\frac{H_{k}\left(x\right)H_{k}\left(a\right)}{2^{k}k!}.$
ⓘ Symbols: $\delta\left(\NVar{x-a}\right)$ : Dirac delta (or Dirac delta function), $H_{\NVar{n}}\left(\NVar{x}\right)$ : Hermite polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ) and $k$ : integer Referenced by: §1.17(iii), §1.17(iv) Permalink: http://dlmf.nist.gov/1.17.E24 Encodings: TeX, pMML, png See also: Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

Spherical Harmonics (§14.30)

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Keywords:: Dirac delta, series representations, spherical harmonics
See also:: Annotations for §1.17(iii), §1.17 and Ch.1

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

ⓘ

Keywords:: Dirac delta, mathematical definitions
Referenced by:: §1.17(ii), Erratum (V1.2.0) §1.17
Permalink:: http://dlmf.nist.gov/1.17.iv
Modification (effective with 1.2.0):: This subsection was almost completely rewritten.
See also:: Annotations for §1.17 and Ch.1

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).