Dirac%20delta%20distribution

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1: 1.16 Distributions

… ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►

§1.16(iii) Dirac Delta Distribution

… ►The Dirac delta distribution is singular. … ►Since $\sqrt{2\pi }ℱ\left(\delta \right)=1$, we have … ► …

2: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

… ►In applications in physics, engineering, and applied mathematics, (see Friedman (1990)), the Dirac delta distribution (§1.16(iii)) is historically and customarily replaced by the Dirac delta (or Dirac delta function) $\delta \left(x\right)$. … ►

Sine and Cosine Functions

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Coulomb Functions (§33.14(iv))

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Airy Functions (§9.2)

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3: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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Morris (1979) gives rational approximations for ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $0.5\le x\le 1$. Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

4: 25.12 Polylogarithms

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

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§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms … ►For a uniform asymptotic approximation for $F_s(x)$ see Temme and Olde Daalhuis (1990).

5: 20 Theta Functions

Chapter 20 Theta Functions

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6: 25.17 Physical Applications

… ►Analogies exist between the distribution of the zeros of $\zeta \left(s\right)$ on the critical line and of semiclassical quantum eigenvalues. … ►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)). …

7: Bibliography P

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Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Evaluation of the Fermi-Dirac integral of half-integer order. Zastos. Mat. 21 (2), pp. 289–301.

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External Links:

ISSN 0044-1899, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§25.18(i).

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The exact distribution of the Wald statistic. Econometrica 54 (4), pp. 881–895.

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External Links:

ISSN 0012-9682, FullText, MathReview (Sadanori Konishi), ZentralBlatt

Referenced by:

§35.9.

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Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

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External Links:

Document

Referenced by:

§25.18(i).

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Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

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8: Bibliography C

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Ellipsoidal distributions of charge or mass. J. Mathematical Phys. 2, pp. 441–450.

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External Links:

MathReview (H. Bremekamp), ZentralBlatt

Referenced by:

§19.33(iv), §19.37(iv).

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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).

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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.

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

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

Referenced by:

§13.29(v), §15.19(v).

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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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Document

Referenced by:

5th item.

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9: 14.30 Spherical and Spheroidal Harmonics

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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

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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

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Distributional Completeness

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

10: Bibliography L

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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).

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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.

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Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.

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Notes:

Single-precision Fortran, maximum accuracy 8S. For Remarks see same journal 13(1970), pp. 124 and 449.

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ISSN 0001-0782, Document, Remark 1 and Remark 2

Referenced by:

§8.28(iv).

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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).

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Sur la distribution des nombres premiers. Comptes Rendus de l’Academie des Sciences, Paris 158, pp. 1869–1872 (French).

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External Links:

ZentralBlatt

Referenced by:

§27.12.

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