Dirac delta function

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11: 9.11 Products

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

12: 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. …

13: 10.22 Integrals

… ►See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. …

14: 14.18 Sums

§14.18 Sums

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§14.18(ii) Addition Theorems

… ►For a series representation of the Dirac delta in terms of products of Legendre polynomials see (1.17.22). … ►

15: 20.13 Physical Applications

§20.13 Physical Applications

… ►with $\kappa =-\mathrm{i}\pi /4$. … ►is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at $t=0$. …Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19). … ►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

16: 18.1 Notation

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§18.1(ii) Main Functions

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Racah: $R_n(x;\alpha ,\beta ,\gamma ,\delta )$.

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Dual Hahn: $R_n(x;\gamma ,\delta ,N)$.

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$q$-Racah: $R_n(x;\alpha ,\beta ,\gamma ,\delta |q)$.

… ►In Koekoek et al. (2010) ${\delta }_x$ denotes the operator $\mathrm{i}{\delta }_x$.

17: 14.30 Spherical and Spheroidal Harmonics

§14.30 Spherical and Spheroidal Harmonics

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Herglotz generating function

►The following is the Herglotz generating function … ►For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). … ►

18: Bibliography C

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The hypergeometric function and the $R$-function near their branch points. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 63–89.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§19.27(vi).

Encodings:

BibTeX

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

Encodings:

BibTeX

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A compact mathematical function package. Australian Computer Journal 16 (3), pp. 107–114.

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Notes:

Includes Pascal function evaluation package

External Links:

ISSN 0004-8917, ZentralBlatt

Referenced by:

§16.27(ii).

Encodings:

BibTeX

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Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha \cdot \alpha +1}{1\cdot 2}\cdot \frac{\beta \cdot \beta +1}{\gamma \cdot \gamma +1}{x}^2+$ etc. ein Quadrat von der Form $z=1+\frac{{\alpha }^{\prime }}{1}\cdot \frac{{\beta }^{\prime }}{{\gamma }^{\prime }}\cdot \frac{{\delta }^{\prime }}{{ϵ}^{\prime }}x+\frac{{\alpha }^{\prime }\cdot {\alpha }^{\prime }+1}{1\cdot 2}\cdot \frac{{\beta }^{\prime }\cdot {\beta }^{\prime }+1}{{\gamma }^{\prime }\cdot {\gamma }^{\prime }+1}\cdot \frac{{\delta }^{\prime }\cdot {\delta }^{\prime }+1}{{ϵ}^{\prime }\cdot {ϵ}^{\prime }+1}{x}^2+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

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External Links:

Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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Numerical evaluation of the Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 71, pp. 677–699.

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Notes:

Double-precision Fortran, maximum precision 12D.

External Links:

ISSN 0067-0049, Document, Code

Referenced by:

§25.12(iii), §25.18(i), 3rd item, 3rd item.

Encodings:

BibTeX

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19: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are quantum wave-functions whose absolute values squared give the probability density of finding the single particle at hand at position $x$ in the $n$th eigenstate, namely that probability is $P(x\to x+\mathrm{\Delta }x)$ = ${|{\psi }_n(x)|}^2\mathrm{\Delta }(x)$, $\mathrm{\Delta }(x)$ being a localized interval on the $x$-axis. … ►

c) Spherical Radial Coulomb Wave Functions

… ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. … ►The bound state $L^2$ eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the $\delta$-function normalized (non-$L^2$) in Chapter 33, where the solutions appear as Whittaker functions. … ►With $N\to \mathrm{\infty }$ the functions normalized as $\delta (ϵ-{ϵ}^{\prime })$ with measure $dr$ are, formally, …