special%20distributions

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1: Foreword

… ►That 1046-page tome proved to be an invaluable reference for the many scientists and engineers who use the special functions of applied mathematics in their day-to-day work, so much so that it became the most widely distributed and most highly cited NIST publication in the first 100 years of the institution’s existence. The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. … ►However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. … ►November 20, 2009 …

2: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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Document

Referenced by:

§33.22(iv).

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On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (4), pp. 1119–1178.

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ISSN 0894-0347, FullText, MathReview (David J. Aldous), ZentralBlatt

Referenced by:

§32.14.

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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Document

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§24.18.

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

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1st item, 4th item.

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Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

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ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt

Referenced by:

§35.10, §35.9.

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3: 3.8 Nonlinear Equations

… ►For fixed-point methods for computing zeros of special functions, see Segura (2002), Gil and Segura (2003), and Gil et al. (2007a, Chapter 7). For describing the distribution of complex zeros of solutions of linear homogeneous second-order differential equations by methods based on the Liouville–Green (WKB) approximation, see Segura (2013). … ►

3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

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Permalink:

http://dlmf.nist.gov/3.8.E15

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Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

4: 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.

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

Referenced by:

§35.9.

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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.

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ISSN 0010-4655, Document, Code

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6th item.

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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)

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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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5: 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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MathReview, ZentralBlatt

Referenced by:

§31.17(ii).

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

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

Referenced by:

§16.23(iii).

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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.

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ISSN 0098-3500, Document, GAMS (module 737; package TOMS), ZentralBlatt

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1st item.

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

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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

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1st item.

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6: 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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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

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

Referenced by:

Ch.13, Ch.15.

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

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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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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)

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

Referenced by:

§10.21(vi).

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

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. 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 $\log$.

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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. …

8: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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

Referenced by:

§28.8(iv).

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

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3rd item.

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

Referenced by:

§18.35(i).

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9: Bibliography D

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The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

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ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

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ISSN 0001-0782, Document

Referenced by:

§8.28(iv).

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Special Functions. World Scientific Publishing Co., Inc., River Edge, NJ.

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External Links:

ISBN 981-02-4393-6, Document, Link, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

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10: 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.

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FullText, Code, ZentralBlatt

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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).

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