special distributions

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21: Preface

… ►Boisvert and Clark were responsible for advising and assisting in matters related to the use of information technology and applications of special functions in the physical sciences (and elsewhere); they also participated in the resolution of major administrative problems when they arose. … ►Miller was responsible for information architecture, specializing LaTeX for the needs of the project, translation from LaTeX to MathML, and the search interface. … ►The project was funded in part by NSF Award 9980036, administered by the NSF’s Knowledge and Distributed Intelligence Program. …

22: Bibliography P

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Handbook of the Normal Distribution. Statistics: Textbooks and Monographs, Vol. 40, Marcel Dekker Inc., New York.

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Notes:

A revised and expanded 2nd edition was published in 1996.

External Links:

ISBN 0-8247-1541-1, MathReview (S. Kotz), ZentralBlatt

Referenced by:

§7.20(iii).

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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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A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:

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

External Links:

ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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Integrals and Series: Special Functions, Vol. 2. Gordon & Breach Science Publishers, New York.

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Notes:

A second edition was published in 1988. Table errata: Math. Comp. v. 65 (1996), no. 215, pp. 1382–1383

External Links:

ISBN 2-88124-097-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.23(iv), §10.43(v), §10.43(vi), §10.44(iv), §10.60(iv), §10.71(iii), §12.12, §12.13(ii), §18.17(ix), §18.18(ix), §25.7, §25.8, §5.13, §6.14(iii), §6.15, §7.14(iii), §7.15, §8.14, §8.15.

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Integrals and Series: More Special Functions, Vol. 3. Gordon and Breach Science Publishers, New York.

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Notes:

Translated from the Russian by G. G. Gould. Table erratum Math. Comp. v. 65 (1996), no. 215, p. 1384.

External Links:

ISBN 2-88124-682-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §11.10(x), §11.10(x), §11.7(v), §11.7(v), §11.9(iv), §13.10(vi), §13.11, §13.23(v), §13.24(ii), (13.6.11_1), (13.6.11_2), §14.17(i), §14.17(iv), §14.18(iv), §14.20(x), §15.14, §15.15, §15.4(i), §16.10, §16.20, §16.20, §16.5, §19.13(i), §19.13(i), §19.13(ii), §20.10(iii), §23.14, §24.13(iii), §24.14(iii), §25.11(ix), §25.11(xi), §25.12(ii), §28.10(iii), §28.28(v), §9.10(ix), §9.11(v).

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23: Bibliography W

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The distribution of the zeros of Jacobian elliptic functions with respect to the parameter $k$ . Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:

ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt

Referenced by:

§22.4(i).

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Special Functions. World Scientific Publishing Co. Inc., Singapore.

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Notes:

Translated from the Chinese by Guo and X. J. Xia

External Links:

ISBN 9971-50-659-9, MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

Ch.13, Ch.15.

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Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions. In Algorithms for Approximation, A. Iske and J. Levesley (Eds.), pp. 331–348.

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External Links:

ISBN 3-540-33283-9, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(i).

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The generalised product moment distribution in samples from a normal multivariate population. Biometrika 20A, pp. 32–52.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

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Error bounds for asymptotic approximations of special functions. Ann. Numer. Math. 2 (1-4), pp. 181–197.

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Notes:

Special functions (Torino, 1993)

External Links:

ISSN 1021-2655, MathReview (Andrei Martínez Finkelshtein), ZentralBlatt

Referenced by:

§10.21(vi).

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24: Bibliography H

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Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).

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Notes:

Reprinted in Oeuvres de Jacques Hadamard. Tomes I, pp. 189–210, Éditions du Centre National de la Recherche Scientifique, Paris, 1968.

External Links:

ISSN 0037-9484, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§25.10(i), §27.2(i).

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Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. 2nd edition, Noordhoff International Publishing, Leyden.

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Notes:

The first edition was published in 1965 by Prentice-Hall. The 2nd edition was reissued in 1983 by Springer, and in 1986 by Martinus Nijhoff Publishers (Kluwer Academic Publishers Group).

External Links:

MathReview (Chia-Shun Yih), ZentralBlatt

Referenced by:

§10.73(i).

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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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External Links:

FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

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Applied and Computational Complex Analysis. Vol. 2: Special Functions—Integral Transforms—Asymptotics—Continued Fractions. Wiley-Interscience [John Wiley & Sons], New York.

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Notes:

Reprinted in 1991.

External Links:

ISBN 0-471-01525-3, MathReview (A. E. Heins), ZentralBlatt

Referenced by:

§1.11(v), Ch.3.

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Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

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Notes:

Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)

External Links:

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

Referenced by:

§8.28(iv).

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25: 27.2 Functions

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27.2.1 $$n=\prod _{r=1}^{\nu \left(n\right)}{p}_r^{a_r},$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Referenced by:

§27.2(i), §27.3

Permalink:

http://dlmf.nist.gov/27.2.E1

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►Tables of primes (§27.21) reveal great irregularity in their distribution. … ►

27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

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Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. …

26: 29.3 Definitions and Basic Properties

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§29.3(ii) Distribution

… ►For the special case $k=k^{\prime }=1/\sqrt{2}$ see Erdélyi et al. (1955, §15.5.2). …

27: 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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Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, ZentralBlatt

Referenced by:

2nd item.

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Statistics on Special Manifolds. Lecture Notes in Statistics, Vol. 174, Springer-Verlag, New York.

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External Links:

ISBN 0-387-00160-3, MathReview (M. Hušková), ZentralBlatt

Referenced by:

§35.9.

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Calculation of angular distributions in complex angular momentum theories of elastic scattering. Molecular Physics 37 (6), pp. 1703–1712.

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External Links:

Document

Referenced by:

§14.31(iii).

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Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.4(i), §35.4(ii).

Encodings:

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28: Bibliography S

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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External Links:

ISSN 1095-7170, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(v).

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Computing the complex zeros of special functions. Numer. Math. 124 (4), pp. 723–752.

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External Links:

ISSN 0029-599X, Document, FullText, MathReview (Miodrag Petković), ZentralBlatt

Referenced by:

§10.21(ix), §10.21(ix), §10.74(vi), §10.74(vi), §3.8(v), §3.8(v).

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On integral representations for Lamé and other special functions. SIAM J. Math. Anal. 11 (4), pp. 702–723.

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External Links:

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

Referenced by:

§29.17(i), §29.8.

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A Guide to Distribution Theory and Fourier Transforms. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL.

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External Links:

ISBN 0-8493-8273-4, MathReview (J. Horváth), ZentralBlatt

Referenced by:

§18.17(v).

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On certain special sets of orthogonal polynomials. Proc. Amer. Math. Soc. 1, pp. 731–737.

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External Links:

ISSN 0002-9939, Document, Link, MathReview (A. C. Offord)

Referenced by:

§18.35(i).

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29: Bibliography O

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

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On the distribution of spacings between zeros of the zeta function. Math. Comp. 48 (177), pp. 273–308.

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External Links:

ISSN 0025-5718, FullText, Author’s Website, MathReview (S. L. Segal), ZentralBlatt

Referenced by:

§25.18(ii).

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Asymptotics and Special Functions. A. K. Peters, Wellesley, MA.

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Notes:

Reprint, with corrections, of original Academic Press edition, 1974

External Links:

ISBN 1-56881-069-5, MathReview, ZentralBlatt

Referenced by:

§1.10(iv), §1.13(viii), §1.13(i), §1.13(ii), §1.13(iv), §1.17(i), §1.4(v), §1.4(ii), §1.4(v), §1.8(i), §1.9(vi), §10.12, §10.14, §10.15, §10.16, §10.17(i), §10.17(iii), §10.17(iv), §10.19(ii), §10.19(ii), §10.2(ii), §10.20(i), §10.20(ii), §10.20(ii), §10.21(i), §10.21(vi), §10.25(ii), §10.27, §10.31, §10.34, §10.37, §10.4, §10.40(i), §10.40(ii), §10.40(iii), §10.40(iii), §10.41(ii), §10.41(iii), §10.41(iv), §10.41(v), §10.42, §10.6(i), §10.8, §10.9(i), §10.9(iii), Ch.10, §11.10(i), (11.11.10), (11.11.11), (11.11.14), (11.11.15), (11.11.16), (11.11.17), (11.11.18), (11.11.19), (11.11.8), §11.11(iii), §11.2(i), §11.2(ii), §11.2(iii), §11.6(i), §11.9(iii), Ch.11, §12.10(vi), §12.2(i), Ch.12, §13.1, §13.14(i), §13.14(ii), §13.14(v), §13.2(i), §13.2(v), §13.20(i), §13.20(ii), §13.20(v), §13.21(i), §13.21(iv), §13.22, §13.7(i), §13.8(i), Ch.13, §14.1, §14.12(ii), §14.15(iii), §14.15(iii), §14.15(iii), §14.16(ii), §14.17(iii), §14.18(ii), §14.2(i), §14.2(ii), §14.2(iii), §14.2(iv), §14.20(vii), §14.20(vii), §14.21(i), §14.21(ii), §14.21(iii), §14.23, §14.24, §14.25, §14.25, §14.26, §14.28(ii), §14.3(ii), §14.3(i), §14.3(ii), §14.3(iii), §14.3(iv), §14.7(i), §14.7(ii), §14.7(iv), §14.8(i), §14.8(iii), §14.9(i), §14.9(ii), §14.9(iii), §14.9(iv), Ch.14, §15.1, §15.10(i), §15.11(i), §15.11(ii), §15.12(iii), §15.2(ii), §15.8(i), Ch.15, §16.2(v), §18.15(iii), §18.15(v), §18.16(v), §2.1(i), §2.1(ii), §2.1(iii), §2.1(iii), §2.1(v), §2.10(ii), §2.10(iii), §2.10(iv), §2.10(i), §2.10(ii), §2.10(iii), §2.10(iv), §2.10(iv), §2.11(i), §2.11(i), §2.11(vi), §2.11(vi), §2.2, §2.2, §2.3(i), §2.3(ii), §2.3(iii), §2.3(iii), §2.3(v), §2.4(i), §2.4(i), §2.4(ii), §2.4(ii), §2.4(iii), §2.4(iii), §2.4(iv), §2.4(iv), §2.4(v), §2.4(vi), §2.7(iii), §2.7(i), §2.7(i), §2.7(ii), §2.7(ii), §2.7(iii), §2.7(iv), §2.8(i), §2.8(ii), §2.8(ii), §2.8(iii), §2.8(iii), §2.8(iv), §2.8(iv), Ch.2, §24.12(i), §24.17(i), §24.9, §26.7(iv), §3.9(vi), §4.13, §5.11(i), §5.11(i), §5.11(ii), §5.11(iii), §5.12, §5.17, §5.2(i), §5.2(ii), §5.4(i), §5.4(ii), §5.5(i), §5.5(ii), §5.5(iii), §5.7(ii), §5.8, §5.9(i), §5.9(ii), Ch.5, §6.12(i), §6.12(i), §6.2(i), §6.2(ii), §6.4, §6.5, §6.6, Ch.6, §7.12(i), §7.12(iii), §7.2(i), §7.2(ii), §7.2(iii), §7.20(i), §7.5, Ch.7, §8.10, §8.11(i), §8.11(i), §8.2(i), §8.2(ii), Ch.8, (9.10.11), (9.10.12), (9.10.17), (9.10.22), §9.10(iv), §9.10(vi), §9.10(viii), (9.12.1), (9.12.10), (9.12.25), (9.12.27), (9.12.3), (9.12.4), (9.12.5), (9.12.6), (9.12.7), (9.12.8), (9.12.9), §9.12(i), §9.12(iii), §9.12(viii), (9.2.1), (9.2.10), (9.2.11), (9.2.12), (9.2.13), (9.2.14), (9.2.15), (9.2.2), (9.2.3), (9.2.4), (9.2.5), (9.2.6), (9.2.7), (9.2.8), (9.2.9), Table 9.2.1, §9.2(i), §9.2(ii), §9.2(iii), §9.2(iv), §9.2(v), (9.4.1), (9.4.2), (9.4.3), (9.4.4), §9.4, (9.5.1), (9.5.2), (9.5.4), §9.5(i), §9.5(ii), (9.6.10), (9.6.11), (9.6.12), (9.6.13), (9.6.14), (9.6.15), (9.6.16), (9.6.17), (9.6.18), (9.6.19), (9.6.2), (9.6.20), (9.6.3), (9.6.4), (9.6.5), (9.6.6), (9.6.7), (9.6.8), (9.6.9), §9.6(i), §9.6(ii), (9.7.10), (9.7.11), (9.7.12), (9.7.13), (9.7.14), (9.7.15), (9.7.2), (9.7.3), (9.7.4), (9.7.5), (9.7.6), (9.7.7), (9.7.8), (9.7.9), Table 9.7.1, §9.7(i), §9.7(ii), §9.7(iii), §9.7(iv), (9.8.1), (9.8.14), (9.8.18), (9.8.2), (9.8.3), (9.8.4), (9.8.5), (9.8.6), (9.8.7), (9.8.8), §9.8(i), §9.8(i), §9.8(ii), §9.8(iii), (9.9.1), (9.9.2), §9.9(i), §9.9(ii), Ch.9, Profile Frank W. J. Olver, Notations P, Notations Q, Notations V, F. W. J. Olver (1974).

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Distribution of the partition function modulo $m$ . Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:

ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§27.14(v).

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30: 1.4 Calculus of One Variable

… ►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus $C^{\mathrm{\infty }}$, and well defined for all values of these variables; possible exceptions being at boundary points. … ►Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17. …