# 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 DiracDelta
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) $\delta\left(x\right)$. …for a suitably chosen sequence of functions $\delta_{n}\left(x\right)$, $n=1,2,\dots$. …
##### 2: 33.14 Definitions and Basic Properties
The function $s\left(\epsilon,\ell;r\right)$ has the following properties:
33.14.13 $\int_{0}^{\infty}s\left(\epsilon_{1},\ell;r\right)s\left(\epsilon_{2},\ell;r% \right)\mathrm{d}r=\delta\left(\epsilon_{1}-\epsilon_{2}\right),$ $\epsilon_{1},\epsilon_{2}>0$,
##### 3: 2.6 Distributional Methods
2.6.17 ${\left\langle f,\phi\right\rangle}=\sum_{s=0}^{n-1}a_{s}\left\langle t^{-s-% \alpha},\phi\right\rangle-\sum_{s=1}^{n}c_{s}\left\langle{\delta^{(s-1)}},\phi% \right\rangle+\left\langle f_{n},\phi\right\rangle$
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 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}{\delta^{(s-1)}}+f_{n}.$
##### 4: Bibliography L
• 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.
• ##### 5: 1.16 Distributions
1.16.40 $\int^{\infty}_{-\infty}\delta\left(t\right){\mathrm{e}^{\mathrm{i}xt}}\mathrm{% d}t=1;$
1.16.43 $\frac{1}{2\pi}\int^{\infty}_{-\infty}{\mathrm{e}^{\mathrm{i}xt}}\mathrm{d}t=% \delta\left(x\right);$
##### 6: 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). …
##### 7: 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. …
##### 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: 33.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $k,\ell$ nonnegative integers. … Dirac delta; see §1.17. …
The main functions treated in this chapter are first the Coulomb radial functions $F_{\ell}\left(\eta,\rho\right)$, $G_{\ell}\left(\eta,\rho\right)$, ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ (Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions $f\left(\epsilon,\ell;r\right)$, $h\left(\epsilon,\ell;r\right)$, $s\left(\epsilon,\ell;r\right)$, $c\left(\epsilon,\ell;r\right)$ (Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. …
• Greene et al. (1979):

$f^{(0)}(\epsilon,\ell;r)=f\left(\epsilon,\ell;r\right)$, $f(\epsilon,\ell;r)=s\left(\epsilon,\ell;r\right)$, $g(\epsilon,\ell;r)=c\left(\epsilon,\ell;r\right)$.

• ##### 10: 9.11 Products
For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). …