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21: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … ►

27 Functions of Number Theory

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22: 9.7 Asymptotic Expansions

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9.7.3 $$\chi (x)\equiv {\pi }^{1/2}\mathrm{\Gamma }\left(\frac{1}{2}x+1\right)/\mathrm{\Gamma }\left(\frac{1}{2}x+\frac{1}{2}\right).$$

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Defines:

$\chi (x)$: function (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\equiv$: equals by definition

Source:

Olver (1997b, (13.02), p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

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9.7.4 $$\chi (x)\sim {(\frac{1}{2}\pi x)}^{1/2}.$$

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi (x)$: function

Source:

Olver (1997b, p. 225)

Referenced by:

Erratum (V1.0.17) for Equations (9.7.3), (9.7.4)

Permalink:

http://dlmf.nist.gov/9.7.E4

Encodings:

pMML, png, TeX

Modification (effective with 1.0.17):

Originally the formula was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

See also:

Annotations for §9.7(i), §9.7 and Ch.9

►Numerical values of $\chi (n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D. ►

Table 9.7.1: $\chi (n)$.

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$n$	$\chi (n)$	$n$	$\chi (n)$	$n$	$\chi (n)$	$n$	$\chi (n)$
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5	2.95	10	4.06	15	4.94	20	5.68

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23: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. …

24: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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Faster computation of Bernoulli numbers. J. Algorithms 13 (3), pp. 431–445.

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External Links:

ISSN 0196-6774, MathReview (P. K. Subramanian), ZentralBlatt

Referenced by:

§24.19(i).

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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25: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. It is also the number of $k$-dimensional lattice paths from $(0,0,\mathrm{\dots },0)$ to $(n_1,n_2,\mathrm{\dots },n_k)$. … ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: ►

26.4.7 $$M_2=\frac{n!}{1^{a_1}(a_1!){\mathrm{\hspace{0.17em}2}}^{a_2}(a_2!)\mathrm{\cdots }{n}^{a_n}(a_n!)}.$$

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Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $a_n$: nonnegative integers and $M_2$: number of permutations

Referenced by:

§26.13

Permalink:

http://dlmf.nist.gov/26.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(i), §26.4 and Ch.26

… ► $M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: …

26: Bibliography K

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Poly-Bernoulli numbers. J. Théor. Nombres Bordeaux 9 (1), pp. 221–228.

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External Links:

ISSN 1246-7405, PostScript, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

1st item.

Encodings:

BibTeX

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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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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.

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External Links:

ISSN 0266-5611, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.11(iv).

Encodings:

BibTeX

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The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.

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External Links:

ISSN 1382-4090, Document, FullText, MathReview (José Luis López), ZentralBlatt

Referenced by:

§18.26(v), §18.26(v).

Encodings:

BibTeX

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27: 26.3 Lattice Paths: Binomial Coefficients

… ► $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$ is the number of ways of choosing $n$ objects from a collection of $m$ distinct objects without regard to order. $\left(\genfrac{}{}{0.0pt}{}{m+n}{n}\right)$ is the number of lattice paths from $(0,0)$ to $(m,n)$. …The number of lattice paths from $(0,0)$ to $(m,n)$, $m\le n$, that stay on or above the line $y=x$ is $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)-\left(\genfrac{}{}{0.0pt}{}{m+n}{m-1}\right).$ … ►

Table 26.3.1: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$.

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$m$	$n$
	…
6	1	6	15	20	15	6	1
…

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Table 26.3.2: Binomial coefficients $\left(\genfrac{}{}{0.0pt}{}{m+n}{m}\right)$ for lattice paths.

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$m$	$n$
	…
3	1	4	10	20	35	56	84	120	165
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28: Bibliography N

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The Development of Prime Number Theory: From Euclid to Hardy and Littlewood. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-66289-8, MathReview (D. R. Heath-Brown), ZentralBlatt

Referenced by:

§27.12.

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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Notes:

References and links to software for factorization and primality testing.

External Links:

Home Page

Referenced by:

6th item.

Encodings:

BibTeX

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29: Bibliography M

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Hill’s Equation. Interscience Tracts in Pure and Applied Mathematics, No. 20, Interscience Publishers John Wiley & Sons, New York-London-Sydney.

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Notes:

Reprinted by Dover Publications, Inc., New York, 1979.

External Links:

MathReview (H. Hochstadt), ZentralBlatt

Referenced by:

§1.3(iii), §28.29(i), §28.29(ii), §28.29(ii), §28.29(iii), §28.29(iii), §28.29(iii), §28.30(i), Ch.28, §29.3(i), §29.9.

Encodings:

BibTeX

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Calculation of the modified Bessel functions of the second kind with complex argument. Math. Comp. 20 (95), pp. 407–412.

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External Links:

ISSN 0025-5718, FullText, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(iii).

Encodings:

BibTeX

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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External Links:

FullText

Referenced by:

§8.24(i).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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30: Bibliography V

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Some Wonderful Formulas $\mathrm{\dots }$ an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.

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External Links:

MathReview (F. Beukers), ZentralBlatt

Referenced by:

(25.6.10), (25.6.8), (25.6.9), §25.6(i), §25.6.

Encodings:

BibTeX

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Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).

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External Links:

ZentralBlatt

Referenced by:

§27.13(ii).

Encodings:

BibTeX

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

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RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions. Comput. Phys. Comm. 92 (2-3), pp. 252–266.

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Notes:

Double-precision Fortran, maximum accuracy 16S. For corrections see same journal 117 (1999), p. 290

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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