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21: 27.2 Functions

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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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7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
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9	6	3	13	22	10	4	36	35	24	4	48	48	16	10	124
10	4	4	18	23	22	2	24	36	12	9	91	49	42	3	57
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22: Bibliography K

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Hypergeometric expansions of Heun polynomials. SIAM J. Math. Anal. 22 (5), pp. 1450–1459.

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External Links:

ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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Addendum: “Hypergeometric expansions of Heun polynomials”. SIAM J. Math. Anal. 22 (6), pp. 1803.

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External Links:

ISSN 0036-1410, Document, MathReview (Peter A. McCoy), ZentralBlatt

Referenced by:

§31.16(ii).

Encodings:

BibTeX

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Algorithm 763: INTERVAL_ARITHMETIC: A Fortran 90 module for an interval data type. ACM Trans. Math. Software 22 (4), pp. 385–392.

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External Links:

ISSN 0098-3500, Document, GAMS (module 763; package TOMS), ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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Algorithm 877: A subroutine package for cylindrical functions of complex order and nonnegative argument. ACM Trans. Math. Software 34 (4), pp. Art. 22, 21.

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External Links:

ISSN 0098-3500, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Equivalence of two graphical calculi. J. Phys. A 25 (22), pp. 6005–6026.

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External Links:

ISSN 0305-4470, Document, MathReview (Niky Kamran), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

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23: 24.2 Definitions and Generating Functions

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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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$22$	$\mathrm{8\hspace{0.33em}54513}$	$138$	$\mathrm{6.19212\hspace{0.33em}3188}$	$\mathrm{\times 10³}$
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Table 24.2.4: Euler numbers $E_n$.

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$n$	$E_n$
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$22$	$-\mathrm{69\hspace{0.33em}34887\hspace{0.33em}43931\hspace{0.33em}37901}$
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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

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	$k$
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$12$	$-\frac{691}{2730}$	$0$	$5$	$0$	$-\frac{33}{2}$	$0$	$22$	$0$	$-\frac{33}{2}$	$0$	$11$	$-6$	$1$
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24: Bibliography T

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Uniform asymptotic expansions of confluent hypergeometric functions. J. Inst. Math. Appl. 22 (2), pp. 215–223.

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External Links:

ISSN 0020-2932, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.8(ii), §13.8(ii).

Encodings:

BibTeX

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Simplified calculation of $\mathrm{Ei}(x)$ for positive arguments, and a short table of $\mathrm{Shi}(x)$ . Math. Comp. 22 (102), pp. 448–449.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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Sul comportamento asintotico dell’$n$-esimo polinomio di Laguerre nell’intorno dell’ascissa $4n$ . Comment. Math. Helv. 22, pp. 150–167.

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External Links:

ISSN 0010-2571, Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§18.16(iv).

Encodings:

BibTeX

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25: 9.9 Zeros

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9.9.14 $${\beta }_k={\mathrm{e}}^{\pi \mathrm{i}/3}T\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.15 $${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)={(-1)}^k\sqrt{2}{\mathrm{e}}^{-\pi \mathrm{i}/6}V\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.16 $${\beta }_k^{\prime }={\mathrm{e}}^{\pi \mathrm{i}/3}U\left(\frac{3}{8}\pi (4k-3)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k^{\prime }$: $k$th complex zero of Airy ${\mathrm{Bi}}^{\prime }$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $U$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.17 $$\mathrm{Bi}\left({\beta }_k^{\prime }\right)={(-1)}^{k-1}\sqrt{2}{\mathrm{e}}^{\pi \mathrm{i}/6}W\left(\frac{3}{8}\pi (4k-3)+\frac{3}{4}\mathrm{i}\ln 2\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k^{\prime }$: $k$th complex zero of Airy ${\mathrm{Bi}}^{\prime }$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $W$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §9.9(iv), §9.9 and Ch.9

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26: 23.9 Laurent and Other Power Series

… ►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\mathrm{\wp }\left(z\right)\to 0$. …

27: 25.5 Integral Representations

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25.5.13 $$\zeta \left(s\right)=\frac{{\pi }^{s/2}}{s(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s\right)}+\frac{{\pi }^{s/2}}{\mathrm{\Gamma }\left(\frac{1}{2}s\right)}{\int }_1^{\mathrm{\infty }}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega (x)}{x}dx,$$ $s\ne 1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

Titchmarsh (1986b, p. 22)

Permalink:

http://dlmf.nist.gov/25.5.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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25.5.14 $$\omega (x)\equiv \sum _{n=1}^{\mathrm{\infty }}{\mathrm{e}}^{-n^2\pi x}=\frac{1}{2}\left({\theta }_3\left(0\right|\mathrm{i}x)-1\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: nonnegative integer and $x$: real variable

Keywords:

definition

Source:

Titchmarsh (1986b, p. 22)

Permalink:

http://dlmf.nist.gov/25.5.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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25.5.17 $$\zeta \left(1+s\right)=\frac{\sin \left(\pi s\right)}{\pi }{\int }_0^{\mathrm{\infty }}\left(\gamma +\psi \left(1+x\right)\right)x^{-s-1}dx,$$

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Symbols:

$\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $\int$: integral, $\sin z$: sine function, $x$: real variable and $s$: complex variable

Keywords:

improper integral, integral representation

Source:

de Bruijn (1937, (22), p. 177)

Permalink:

http://dlmf.nist.gov/25.5.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §25.5(ii), §25.5 and Ch.25

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28: 26.3 Lattice Paths: Binomial Coefficients

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29: 26.7 Set Partitions: Bell Numbers

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30: 28.26 Asymptotic Approximations for Large $q$

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28.26.4 $${\mathrm{Fc}}_m(z,h)\sim 1+\frac{s}{8h{\cosh }^{2}z}+\frac{1}{2^{11}{h}^2}\left(\frac{s^4+86s^2+105}{{\cosh }^{4}z}-\frac{s^4+22s^2+57}{{\cosh }^{2}z}\right)+\frac{1}{2^{14}{h}^3}\left(-\frac{s^5+14s^3+33s}{{\cosh }^{2}z}-\frac{2s^5+124s^3+1122s}{{\cosh }^{4}z}+\frac{3s^5+290s^3+1627s}{{\cosh }^{6}z}\right)+\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\cosh z$: hyperbolic cosine function, $m$: integer, $h$: parameter and $z$: complex variable

A&S Ref:

20.9.9 (in slightly different notation)

Referenced by:

§28.26(i)

Permalink:

http://dlmf.nist.gov/28.26.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.26(i), §28.26 and Ch.28

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