# 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 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) $\delta\left(x\right)$. This is a symbolic function with the properties: …Such a sequence is called a delta sequence and we write, symbolically, …
##### 2: 33.1 Special Notation
 $k,\ell$ nonnegative integers. … Dirac delta; see §1.17. …
##### 3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
of the Dirac delta distribution. …
1.18.30 $\sum_{n=0}^{\infty}\phi_{\lambda_{n}}(x)\overline{\phi_{\lambda_{n}}(y)}=% \delta\left(x-y\right).$
Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta\left(x-y\right)=\frac{1}{\sqrt{xy}}\delta\left(x-y\right)$, … These latter results also correspond to use of the $\delta\left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2). …
##### 4: 1.16 Distributions
###### §1.16(iii) DiracDelta Distribution
The Dirac delta distribution is singular. …
1.16.40 $\int^{\infty}_{-\infty}\delta\left(t\right){\mathrm{e}}^{\mathrm{i}xt}\,% \mathrm{d}t=1;$
1.16.43 $\frac{1}{2\pi}\int^{\infty}_{-\infty}{\mathrm{e}}^{\mathrm{i}xt}\,\mathrm{d}t=% \delta\left(x\right);$
##### 5: 33.14 Definitions and Basic Properties
The function $s\left(\epsilon,\ell;r\right)$ has the following properties:
33.14.13 $\int_{0}^{\infty}s\left(\epsilon_{1},\ell;r\right)s\left(\epsilon_{2},\ell;r% \right)\,\mathrm{d}r=\delta\left(\epsilon_{1}-\epsilon_{2}\right),$ $\epsilon_{1},\epsilon_{2}>0$,
where the right-hand side is the Dirac delta1.17). …
##### 6: 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. … Let $\,\mathrm{d}\alpha(x)=w(x)\,\mathrm{d}x+\sum_{n=1}^{N}w_{n}\delta\left(x-x_{n}% \right)\,\mathrm{d}x$, $x_{n}\in(a,b)$, $n=1,\dots N$. …
##### 7: 18.1 Notation
 $x,y,t$ real variables. … Dirac delta (§1.17). …
##### 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
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 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-1}-\sum_{s=1}^{n}d_{s}{\delta}^{(s-1)}+f_{n}.$
##### 10: 20.13 Physical Applications
is also a solution of (20.13.2), and it approaches a Dirac delta1.17) at $t=0$. …