Dirac delta

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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 and engineering, 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 an operator with the properties: …Such a sequence is called a delta sequence and we write, symbolically, … ►

2: 33.1 Special Notation

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$k,\mathrm{\ell }$	nonnegative integers.
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$\delta \left(x\right)$	Dirac delta; see §1.17.
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3: 2.6 Distributional Methods

… ►The Dirac delta distribution in (2.6.17) is given by ►

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

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function) and $⟨\mathrm{\Lambda },\varphi ⟩$: inner-product of distribution

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

… ►Since $\delta =\mathit{D}H$, it follows that for $\mu \ne 1,2,\mathrm{\dots }$, ►

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

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mu$: order and $\mathrm{\ast }$: convolution

Referenced by:

§2.6(iii)

Permalink:

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

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See also:

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

… ►

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:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $f(t)$: 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

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

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4: 1.16 Distributions

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§1.16(iii) Dirac Delta Distribution

… ►The Dirac delta distribution is singular. … ►

1.16.39 $$ℱ\left(\delta \right)=\frac{1}{\sqrt{2\pi }},$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\delta }_x$: Dirac delta distribution and $ℱ\left(u\right)$: Fourier transform of a tempered distribution

Permalink:

http://dlmf.nist.gov/1.16.E39

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Annotations for §1.16(viii), §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:

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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

…

5: 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

Encodings:

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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

►where the right-hand side is the Dirac delta (§1.17). …

6: 18.36 Miscellaneous Polynomials

… ►These are OP’s on the interval $(-1,1)$ with respect to an orthogonality measure obtained by adding constant multiples of “Dirac delta weights” at $-1$ and $1$ to the weight function for the Jacobi polynomials. …

7: 18.1 Notation

… ► ►►►

$x,y$	real variables.
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$\delta \left(x-a\right)$	Dirac delta (§1.17).
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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: 20.13 Physical Applications

… ►is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at $t=0$. …

10: 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). …