# Dirac delta distribution

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##### 1: 1.16 Distributions
The Dirac delta distribution is singular. …
##### 2: 2.6 Distributional Methods
The Dirac delta distribution in (2.6.17) is given by
2.6.19 $\left\langle{\delta}^{(s)},\phi\right\rangle=(-1)^{s}\phi^{(s)}(0),$ $s=0,1,2,\dots$;
2.6.38 $t^{\mu-1}\ast{\delta}^{(s-1)}=\frac{\Gamma\left(\mu\right)}{\Gamma\left(\mu+1-% s\right)}t^{\mu-s},$ $t>0$.
2.6.41 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}{\delta}^{(s-1)}+f_{n},$
2.6.42 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}{\delta}^{(s-1)}+f_{n}.$
##### 3: 1.17 Integral and Series Representations of the Dirac Delta
In applications in physics, engineering, and applied mathematics, (see Friedman (1990)), the Dirac delta distribution1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) $\delta\left(x\right)$. … 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). …
##### 4: 1.4 Calculus of One Variable
Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17. …
##### 5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
of the Dirac delta distribution. …
##### 6: 14.30 Spherical and Spheroidal Harmonics
14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$
###### Distributional Completeness
For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). …
##### 7: 20.13 Physical Applications
is also a solution of (20.13.2), and it approaches a Dirac delta1.17) at $t=0$. … In the singular limit $\Im\tau\rightarrow 0+$, the functions $\theta_{j}\left(z\middle|\tau\right)$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
##### 8: Bibliography L
• D. A. Levine (1969) Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.
• Y. T. Li and R. Wong (2008) Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.
• J. E. Littlewood (1914) Sur la distribution des nombres premiers. Comptes Rendus de l’Academie des Sciences, Paris 158, pp. 1869–1872 (French).