Dirac%20delta

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11: 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). …

12: 2.6 Distributional Methods

… ►The Dirac delta distribution in (2.6.17) is given by … ►Since $\delta =𝐷H$, it follows that for $\mu \ne 1,2,\mathrm{\dots }$, … ►when $\alpha =1$, where … ►when $\alpha \ne \beta$, where …For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

13: 20 Theta Functions

Chapter 20 Theta Functions

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14: 25.18 Methods of Computation

… ►For dilogarithms and polylogarithms see Jacobs and Lambert (1972), Osácar et al. (1995), Spanier and Oldham (1987, pp. 231–232), and Zudilin (2007). ►For Fermi–Dirac and Bose–Einstein integrals see Cloutman (1989), Gautschi (1993), Mohankumar and Natarajan (1997), Natarajan and Mohankumar (1993), Paszkowski (1988, 1991), Pichon (1989), and Sagar (1991a, b). …

15: 14.30 Spherical and Spheroidal Harmonics

… ►

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

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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

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

16: Bibliography N

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On the numerical evaluation of the generalised Fermi-Dirac integrals. Comput. Phys. Comm. 76 (1), pp. 48–50.

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External Links:

ISSN 0010-4655, Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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17: Bibliography G

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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

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External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Algorithm 745: Computation of the complete and incomplete Fermi-Dirac integral. ACM Trans. Math. Software 21 (3), pp. 221–232.

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Notes:

Double-precision Fortran, maximum accuracy 20D.

External Links:

ISSN 0098-3500, Document, GAMS (module 745; package TOMS), ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

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Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

Encodings:

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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18: 25.17 Physical Applications

… ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …

19: Bibliography L

… ►

Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

Encodings:

BibTeX

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

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

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External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

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20: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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:

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

BibTeX

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

Encodings:

BibTeX

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