integral representations for Dirac delta

SearchAdvancedHelp

(0.004 seconds)

8 matching pages

1: 1.17 Integral and Series Representations of the Dirac Delta

… ►

§1.17(ii) Integral Representations

… ►

Sine and Cosine Functions

… ►

Bessel Functions and Spherical Bessel Functions (§§10.2(ii), 10.47(ii))

… ►

Coulomb Functions (§33.14(iv))

… ►

Airy Functions (§9.2)

…

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 $s(ϵ,\mathrm{\ell };r)$ has the following properties: …

4: 9.11 Products

… ►For an integral representation of the Dirac delta involving a product of two $\mathrm{Ai}$ functions see §1.17(ii). …

5: Bibliography L

… ►

Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

ⓘ

External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

…

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 $[0,\mathrm{\infty })$. … ►For fixed angular momentum $\mathrm{\ell }$ the appropriate self-adjoint extension of the above operator may have both a discrete spectrum of negative eigenvalues ${\lambda }_n,n=0,1,\mathrm{\dots },N-1$, with corresponding $L^2([0,\mathrm{\infty }),r^{2}dr)$ eigenfunctions ${\varphi }_n(r)$, and also a continuous spectrum $\lambda \in [0,\mathrm{\infty })$, with Dirac-delta normalized eigenfunctions ${\varphi }_{\lambda }(r)$, also with measure $r^{2}dr$. …

8: 14.30 Spherical and Spheroidal Harmonics

… ►

Explicit Representation

… ►

14.30.8 $${\int }_0^{2\pi }{\int }_0^{\pi }\overline{Y_{l_1,m_1}(\theta ,\varphi )}{Y}_{l_2,m_2}(\theta ,\varphi )\sin \theta d\theta d\varphi ={\delta }_{l_1,l_2}{\delta }_{m_1,m_2}.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer and $l$: nonnegative integer

Referenced by:

§14.30(ii)

Permalink:

http://dlmf.nist.gov/14.30.E8

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.

See also:

Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

►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). …