# 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 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
 $k,\ell$ nonnegative integers. … Dirac delta; see §1.17. …
##### 3: 2.6 Distributional Methods
The Dirac delta distribution in (2.6.17) is given by
2.6.19 $\left\langle{\delta^{(s)}},\phi\right\rangle=(-1)^{s}\phi^{(s)}(0),$ $s=0,1,2,\dots$;
Since $\delta=DH$, it follows that for $\mu\neq 1,2,\dots$,
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 $f=\sum_{s=0}^{n-1}a_{s}t^{-s-\alpha}-\sum_{s=1}^{n}c_{s}{\delta^{(s-1)}}+f_{n},$
##### 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: 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. … 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: 20.13 Physical Applications
is also a solution of (20.13.2), and it approaches a Dirac delta1.17) 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). …