# integral representations for Dirac delta

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##### 2: 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). …
##### 3: 33.14 Definitions and Basic Properties
The function $s\left(\epsilon,\ell;r\right)$ has the following properties: …
##### 4: 9.11 Products
For an integral representation of the Dirac delta involving a product of two $\operatorname{Ai}$ functions see §1.17(ii). …
##### 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: 10.22 Integrals
See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. …
##### 7: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
of the Dirac delta distribution. … 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). … Thus, and this is a case where $q(x)$ is not continuous, if $q(x)=-\alpha\delta\left(x-a\right)$, $\alpha>0$, there will be an $L^{2}$ eigenfunction localized in the vicinity of $x=a$, with a negative eigenvalue, thus disjoint from the continuous spectrum on $[0,\infty)$. … For fixed angular momentum $\ell$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues $\lambda_{n},n=0,1,\dots,N-1$, with corresponding $L^{2}\left([0,\infty),r^{2}\,\mathrm{d}r\right)$ eigenfunctions $\phi_{n}(r)$, and also a continuous spectrum $\lambda\in[0,\infty)$, with Dirac-delta normalized eigenfunctions $\phi_{\lambda}(r)$, also with measure $r^{2}\,\mathrm{d}r$. …
##### 8: 14.30 Spherical and Spheroidal Harmonics
###### Explicit Representation
14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$
See also (34.3.22), and for further related integrals see Askey et al. (1986). … For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). …