Dirac delta function

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1: 1.17 Integral and Series Representations of the Dirac Delta

§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) $\delta \left(x\right)$. …for a suitably chosen sequence of functions ${\delta }_n\left(x\right)$, $n=1,2,\mathrm{\dots }$. … ►

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

… ►

Airy Functions (§9.2)

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2: 33.14 Definitions and Basic Properties

… ►The function $s(ϵ,\mathrm{\ell };r)$ has the following properties: ►

33.14.13 $${\int }_0^{\mathrm{\infty }}s({ϵ}_1,\mathrm{\ell };r)s({ϵ}_2,\mathrm{\ell };r)dr=\delta \left({ϵ}_1-{ϵ}_2\right),$$ ${ϵ}_1,{ϵ}_2>0$,

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Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E13

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Addition (effective with 1.0.24):

The constraint ${ϵ}_1,{ϵ}_2>0$ was added.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

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3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

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1.18.19 $$\delta \left(x-y\right)=\sum _{n=0}^{\mathrm{\infty }}{\varphi }_n(x)\overline{{\varphi }_n(y)},$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(vi), §1.18(ii)

Permalink:

http://dlmf.nist.gov/1.18.E19

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Annotations for §1.18(ii), §1.18 and Ch.1

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1.18.30 $$\sum _{n=0}^{\mathrm{\infty }}{\varphi }_{{\lambda }_n}(x)\overline{{\varphi }_{{\lambda }_n}(y)}=\delta \left(x-y\right).$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate and $n$: nonnegative integer

Referenced by:

§1.18(v), §1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E30

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Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

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1.18.39 $$\delta \left(x-y\right)=\frac{1}{\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}n(x-y)}.$$

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Symbols:

$\delta \left(x-a\right)$: 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 $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/1.18.E39

Encodings:

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Annotations for §1.18(v), §1.18(v), §1.18 and Ch.1

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1.18.45 $$\delta \left(x-y\right)={\int }_0^{\mathrm{\infty }}{\varphi }_{\lambda }(x)\overline{{\varphi }_{\lambda }(y)}d\lambda .$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $dx$: differential of $x$ and $\int$: integral

Referenced by:

§1.18(viii)

Permalink:

http://dlmf.nist.gov/1.18.E45

Encodings:

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See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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1.18.56 $$\delta \left(x-y\right)={\int }_0^{\mathrm{\infty }}\sqrt{xt}{J}_{\nu }\left(xt\right)\sqrt{yt}{J}_{\nu }\left(yt\right)dt,$$ $\mathrm{\Re }\nu >-1$, $x,y\ge 0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral and $\mathrm{\Re }$: real part

Permalink:

http://dlmf.nist.gov/1.18.E56

Encodings:

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See also:

Annotations for §1.18(vi), §1.18(vi), §1.18 and Ch.1

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4: 1.4 Calculus of One Variable

… ►If, for example, $\alpha (x)=H\left(x-x_n\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $d\alpha (x)$ is $\delta \left(x-x_n\right)dx$, where $\delta \left(x-x_n\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, ${\int }_a^{b}f(x)d\alpha (x)=f(x_n)$ for $x_n\in (a,b)$ and $0$ otherwise. Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17. …

5: Bibliography L

… ►

Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

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External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

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6: 1.16 Distributions

… ►

1.16.15 $$𝐷H=\delta ,$$

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Symbols:

$H\left(x\right)$: Heaviside function, ${\delta }_x$: Dirac delta distribution and $𝐷f$: distributional derivative

Referenced by:

§1.16(viii)

Permalink:

http://dlmf.nist.gov/1.16.E15

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See also:

Annotations for §1.16(iv), §1.16 and Ch.1

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1.16.16 $$𝐷H\left(x-x_0\right)={\delta }_{x_0}.$$

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Symbols:

$H\left(x\right)$: Heaviside function, ${\delta }_x$: Dirac delta distribution and $𝐷f$: distributional derivative

Referenced by:

§1.16(iv), §1.16(iv)

Permalink:

http://dlmf.nist.gov/1.16.E16

Encodings:

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See also:

Annotations for §1.16(iv), §1.16 and Ch.1

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1.16.40 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(t\right){\mathrm{e}}^{\mathrm{i}xt}dt=1;$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.16.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

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1.16.43 $$\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{\mathrm{i}xt}dt=\delta \left(x\right);$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.16.E43

Encodings:

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See also:

Annotations for §1.16(viii), §1.16 and Ch.1

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1.16.45 $${\mathrm{sign}}^{\prime }=2H^{\prime }=2\delta ,$$

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Symbols:

$H\left(x\right)$: Heaviside function, ${\delta }_x$: Dirac delta distribution and $\mathrm{sign}x$: sign of

Permalink:

http://dlmf.nist.gov/1.16.E45

Encodings:

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See also:

Annotations for §1.16(viii), §1.16 and Ch.1

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7: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

8: 10.59 Integrals

… ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). …

9: 2.6 Distributional Methods

… ►

2.6.17 $$⟨f,\varphi ⟩=\sum _{s=0}^{n-1}{a}_s⟨t^{-s-\alpha },\varphi ⟩-\sum _{s=1}^n{c}_s⟨{\delta }^{(s-1)},\varphi ⟩+⟨f_n,\varphi ⟩$$

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Symbols:

${\delta }_x$: Dirac delta distribution, $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $c\ne 0$: real, $f(t)$: locally integrable function, $a_n$: coefficients, $n$: nonnegative integer and $f_n(t)$: remainder

Referenced by:

§2.6(ii), §2.6(ii), §2.6(ii), §2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E17

Encodings:

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Annotations for §2.6(ii), §2.6 and Ch.2

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2.6.19 $$⟨{\delta }^{(s)},\varphi ⟩={(-1)}^s{\varphi }^{(s)}(0),$$ $s=0,1,2,\mathrm{\dots }$;

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Symbols:

${\delta }_x$: Dirac delta distribution and $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function

Permalink:

http://dlmf.nist.gov/2.6.E19

Encodings:

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See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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2.6.38 $$t^{\mu -1}\ast {\delta }^{(s-1)}=\frac{\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\mu +1-s\right)}{t}^{\mu -s},$$ $t>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_x$: Dirac delta distribution, $\mu$: order and $\ast$: convolution

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E38

Encodings:

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Annotations for §2.6(iii), §2.6 and Ch.2

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2.6.41 $$f=\sum _{s=0}^{n-1}{a}_s{t}^{-s-\alpha }-\sum _{s=1}^n{c}_s{\delta }^{(s-1)}+f_n,$$

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Symbols:

${\delta }_x$: Dirac delta distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients, $f_n(t)$: remainder and $c_s$: coefficients

Permalink:

http://dlmf.nist.gov/2.6.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iii), §2.6 and Ch.2

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2.6.42 $$f=\sum _{s=0}^{n-1}{a}_s{t}^{-s-1}-\sum _{s=1}^n{d}_s{\delta }^{(s-1)}+f_n.$$

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Symbols:

${\delta }_x$: Dirac delta distribution, $d_s$: coefficients, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients and $f_n(t)$: remainder

Permalink:

http://dlmf.nist.gov/2.6.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(iii), §2.6 and Ch.2

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10: 33.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,\mathrm{\ell }$	nonnegative integers.
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$\delta \left(x\right)$	Dirac delta; see §1.17.
…

►The main functions treated in this chapter are first the Coulomb radial functions $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, $c(ϵ,\mathrm{\ell };r)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. … ►

$f^{(0)}(ϵ,\mathrm{\ell };r)=f(ϵ,\mathrm{\ell };r)$, $f(ϵ,\mathrm{\ell };r)=s(ϵ,\mathrm{\ell };r)$, $g(ϵ,\mathrm{\ell };r)=c(ϵ,\mathrm{\ell };r)$.