# 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, 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 $\delta_{n}\left(x\right)$, $n=1,2,\dots$. …
##### 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: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
1.18.30 $\sum_{n=0}^{\infty}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{n}}(y)}=% \delta\left(x-y\right).$
1.18.39 $\delta\left(x-y\right)=\frac{1}{\pi}\sum_{n=-\infty}^{\infty}{\mathrm{e}}^{2% \mathrm{i}n(x-y)}.$
1.18.56 $\delta\left(x-y\right)=\int_{0}^{\infty}\sqrt{xt}J_{\nu}\left(xt\right)\sqrt{% yt}J_{\nu}\left(yt\right)\,\mathrm{d}t,$ $\Re\nu>-1$, $x,y\geq 0$.
##### 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 $\,\mathrm{d}\alpha(x)$ is $\delta\left(x-x_{n}\right)\,\mathrm{d}x$, where $\delta\left(x-x_{n}\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, $\int_{a}^{b}f(x)\,\mathrm{d}\alpha(x)=f(x_{n})$ 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
• 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.
• ##### 6: 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 $\frac{1}{2\pi}\int^{\infty}_{-\infty}{\mathrm{e}}^{\mathrm{i}xt}\,\mathrm{d}t=% \delta\left(x\right);$
##### 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 ${\left\langle f,\phi\right\rangle}=\sum_{s=0}^{n-1}a_{s}\left\langle t^{-s-% \alpha},\phi\right\rangle-\sum_{s=1}^{n}c_{s}\left\langle{\delta}^{(s-1)},\phi% \right\rangle+\left\langle f_{n},\phi\right\rangle$
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 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}{\delta}^{(s-1)}+f_{n}.$
##### 10: 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):

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