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11: 1.1 Special Notation

§1.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►►

$x,y$	real variables.
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$⟨\mathrm{\Lambda },\varphi ⟩$	action of distribution $\mathrm{\Lambda }$ on test function $\varphi$.
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12: DLMF Project News

error generating summary

13: 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) $\delta \left(x\right)$. … ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). … ►(1.17.22)–(1.17.24) are special cases of Morse and Feshbach (1953a, Eq. (6.3.11)). …

14: Notices

… ►Limited copying and internal distribution of the content of these pages is permitted for research and teaching. Reproduction, copying, or distribution for any commercial purpose is strictly prohibited. … ►The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. Using special functions in applications often requires computing them. … ►

In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

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15: 1.2 Elementary Algebra

… ►distributive if $𝐁$ and $𝐂$ have the same dimensions … ►Special cases are the Euclidean length or $l^2$ norm … ►Square $n\times n$ matrices (said to be of order $n$ ) dominate the use of matrices in the DLMF, and they have many special properties. … ►

Special Forms of Square Matrices

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Special Properties and Definitions Relating to Square Matrices

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16: Bibliography K

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Special Functions and Perturbations of Black Holes. In Special Functions (Hong Kong, 1999), pp. 140–151.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Integral equations for special functions of Heun class. Methods Appl. Anal. 3 (4), pp. 447–456.

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External Links:

ISSN 1073-2772, FullText, MathReview, ZentralBlatt

Referenced by:

§31.10(ii).

Encodings:

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Special functions and the Bieberbach conjecture. Amer. Math. Monthly 95 (8), pp. 689–696.

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External Links:

ISSN 0002-9890, FullText, MathReview (S. L. Shukla), ZentralBlatt

Referenced by:

§16.23(iii).

Encodings:

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Overview of some new results concerning the theory and applications of the Rayleigh special function. Comput. Math. Math. Phys. 48 (9), pp. 1454–1507.

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External Links:

ISSN 0965-5425, Document, FullText

Referenced by:

§10.21(xiii), §10.21(xiii).

Encodings:

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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

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Notes:

In cooperation with W. Rudert

External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

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17: 7.20 Mathematical Applications

… ►The spiral has several special properties (see Temme (1996b, p. 184)). … ►The normal distribution function with mean $m$ and standard deviation $\sigma$ is given by …For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

18: Bibliography G

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Quasirandom distributed bases for bound problems. J. Chem. Phys. 114 (9), pp. 3929–3939.

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Referenced by:

§18.39(iii).

Encodings:

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Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-0101-0, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

Encodings:

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Computational Methods in Special Functions – A Survey. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.5.

Encodings:

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

Encodings:

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Computing special functions by using quadrature rules. Numer. Algorithms 33 (1-4), pp. 265–275.

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Notes:

International Conference on Numerical Algorithms, Vol. I (Marrakesh, 2001)

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§3.5(ix).

Encodings:

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19: 16.23 Mathematical Applications

… ►In Janson et al. (1993) limiting distributions are discussed for the sparse connected components of these graphs, and the asymptotics of three ${}_2{}^{}F_2^{}$ functions are applied to compute the expected value of the excess. … ►Many combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. …

20: 14.30 Spherical and Spheroidal Harmonics

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Special Values

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Distributional Completeness

… ►The special class of spherical harmonics $Y_{l,m}(\theta ,\varphi )$, defined by (14.30.1), appear in many physical applications. …