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1: 10.75 Tables

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Makinouchi (1966) tabulates all values of $j_{\nu ,m}$ and $y_{\nu ,m}$ in the interval $(0,100)$, with at least 29S. These are for $\nu =0(1)5$, 10, 20; $\nu =\frac{3}{2}$, $\frac{5}{2}$; $\nu =m/n$ with $m=1(1)n-1$ and $n=3(1)8$, except for $\nu =\frac{1}{2}$.

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2: Bibliography I

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The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

BibTeX

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An integral which occurs in statistics. Proceedings of the Cambridge Philosophical Society 29, pp. 271–276.

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External Links:

Document, ZentralBlatt

Referenced by:

§35.3(i), §35.3(ii).

Encodings:

BibTeX

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

BibTeX

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Centre for Experimental and Constructive Mathematics, Simon Fraser University, Canada.

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Notes:

From a decimal approximation, find the most likely exact value.

External Links:

Inverse Symbolic Calculator

Referenced by:

3rd item.

Encodings:

BibTeX

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3: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►Table 26.4.1 gives numerical values of multinomials and partitions $\lambda ,M_1,M_2,M_3$ for $1\le m\le n\le 5$. …For each $n$ all possible values of $a_1,a_2,\mathrm{\dots },a_n$ are covered. ►

Table 26.4.1: Multinomials and partitions.

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$n$	$m$	$\lambda$	$M_1$	$M_2$	$M_3$
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$5$	$2$	$2^1,3^1$	$10$	$20$	$10$
$5$	$3$	$1^2,3^1$	$20$	$20$	$10$
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4: 25.12 Polylogarithms

… ►Other notations and names for ${\mathrm{Li}}_2\left(z\right)$ include $S_2(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and ${\mathrm{L}}_2(z)$ (Maximon (2003)). ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches. … ►For other values of $z$, ${\mathrm{Li}}_s\left(z\right)$ is defined by analytic continuation. …

5: 27.2 Functions

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

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

Encodings:

pMML, png, TeX

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

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27.2.4 $$p_n\sim n\ln n.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.2.E4

Encodings:

pMML, png, TeX

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

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27.2.14 $$\mathrm{\Lambda }\left(n\right)=\ln p,$$ $n=p^a$,

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Defines:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E14

Encodings:

pMML, png, TeX

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

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Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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3	2	2	4	16	8	5	31	29	28	2	30	42	12	8	96
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6: 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).

Encodings:

BibTeX

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Algorithm 537: Characteristic values of Mathieu’s differential equation. ACM Trans. Math. Software 5 (1), pp. 112–117.

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External Links:

ISSN 0098-3500, Document, GAMS (module 537; package TOMS)

Referenced by:

2nd item, 2nd item.

Encodings:

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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.

Encodings:

BibTeX

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Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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External Links:

ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§30.12.

Encodings:

BibTeX

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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7: 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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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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Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.

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Notes:

An Algol procedure for calculating the complete elliptic integrals of the first and the second kind with complex modulus $k$ is included.

External Links:

ISSN 0029-599X, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.39(ii).

Encodings:

BibTeX

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Stirling numbers of the second kind. Duke Math. J. 25 (1), pp. 29–43.

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External Links:

Document, MathReview (A. Sade), ZentralBlatt

Referenced by:

§26.1, §26.8(vii), Notations S.

Encodings:

BibTeX

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Rational solutions of the second and the fourth Painlevé equations. Funkcial. Ekvac. 28 (1), pp. 1–32.

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External Links:

ISSN 0532-8721, FullText, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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8: Bibliography B

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Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

Encodings:

BibTeX

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Algorithms for evaluating spherical Bessel functions in the complex domain. Zh. Vychisl. Mat. i Mat. Fiz. 28 (12), pp. 1779–1788, 1918.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 6, 122–128

External Links:

ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

Encodings:

BibTeX

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Products of generalized hypergeometric series. Proc. London Math. Soc. (2) 28 (2), pp. 242–254.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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Transformations of generalized hypergeometric series. Proc. London Math. Soc. (2) 29 (2), pp. 495–502.

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External Links:

Document, ZentralBlatt

Referenced by:

§16.6.

Encodings:

BibTeX

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Über Sturm-Liouvillesche Polynomsysteme. Math. Z. 29 (1), pp. 730–736.

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External Links:

ISSN 0025-5874, Link, MathReview Entry, ZentralBlatt

Referenced by:

item 1..

Encodings:

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9: Bibliography O

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

BibTeX

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Bessel Functions. Part III: Zeros and Associated Values. Royal Society Mathematical Tables, Volume 7, Cambridge University Press, Cambridge-New York.

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External Links:

ISBN 0-521-06153-9, MathReview (L. Fox), ZentralBlatt

Referenced by:

§10.21(vi), §10.21(viii), §10.58, §10.74(vi), 1st item, 2nd item.

Encodings:

BibTeX

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Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.

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Notes:

See also Olver (1997b)

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

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External Links:

ISSN 0891-6837, MathReview Entry, ZentralBlatt

Referenced by:

§10.29(i), §10.6(i).

Encodings:

BibTeX

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10: Bibliography K

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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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An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.

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External Links:

ISSN 0044-2275, Document, MathReview, ZentralBlatt

Referenced by:

§10.71(ii).

Encodings:

BibTeX

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Clebsch-Gordan coefficients for $\mathrm{SU}(2)$ and Hahn polynomials. Nieuw Arch. Wisk. (3) 29 (2), pp. 140–155.

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External Links:

ISSN 0028-9825, MathReview (Jiří Patera), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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Zeros of exceptional Hermite polynomials. J. Approx. Theory 200, pp. 28–39.

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External Links:

ISSN 0021-9045, Document, Link, MathReview (Mario Pérez Riera)

Referenced by:

§18.39(i).

Encodings:

BibTeX

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