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11: 23.17 Elementary Properties

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$\lambda \left(\mathrm{i}\right)=\frac{1}{2},$

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23.17.5 $$1728J\left(\tau \right)=q^{-2}+744+\mathrm{1\hspace{0.33em}96884}{q}^2+\mathrm{214\hspace{0.33em}93760}{q}^4+\mathrm{\cdots },$$

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Symbols:

$J\left(\tau \right)$: Klein’s complete invariant, $q$: nome and $\tau$: complex variable

Referenced by:

§23.17(ii)

Permalink:

http://dlmf.nist.gov/23.17.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

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23.17.6 $$\eta \left(\tau \right)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{q}^{{(6n+1)}^2/12}.$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Permalink:

http://dlmf.nist.gov/23.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(ii), §23.17 and Ch.23

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23.17.8 $$\eta \left(\tau \right)=q^{1/12}\prod _{n=1}^{\mathrm{\infty }}(1-q^{2n}),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $q$: nome, $n$: integer and $\tau$: complex variable

Referenced by:

§23.15(ii), §23.19

Permalink:

http://dlmf.nist.gov/23.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §23.17(iii), §23.17 and Ch.23

►with $q^{1/12}={\mathrm{e}}^{\mathrm{i}\pi \tau /12}$.

12: 1.3 Determinants, Linear Operators, and Spectral Expansions

… ►For $n=2$: … ►for every distinct pair of $j,k$, or when one of the factors ${\sum }_{k=1}^n{a}_{jk}^2$ vanishes. … ►where ${\omega }_1,{\omega }_2,\mathrm{\dots },{\omega }_n$ are the $n$th roots of unity (1.11.21). … ►Let $a_{j,k}$ be defined for all integer values of $j$ and $k$, and ${𝐷}_n[a_{j,k}]$ denote the $(2n+1)\times (2n+1)$ determinant … ►Taking $l^2$ norms, …

13: 24.2 Definitions and Generating Functions

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$B_{2n+1}=0$ ,

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${(-1)}^{n+1}{B}_{2n}>0$ , $n=1,2,\mathrm{\dots }$.

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$E_{2n+1}=0$ ,

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${(-1)}^n{E}_{2n}>0$ .

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24.2.9 $$E_n=2^n{E}_n\left(\frac{1}{2}\right)=\text{integer},$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.2

Permalink:

http://dlmf.nist.gov/24.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(ii), §24.2 and Ch.24

…

14: 26.21 Tables

… ►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts $\cancel{\equiv }\pm 2(mod5)$, partitions into parts $\cancel{\equiv }\pm 1(mod5)$, and unrestricted plane partitions up to 100. It also contains a table of Gaussian polynomials up to ${\left[\genfrac{}{}{0.0pt}{}{12}{6}\right]}_q$. …

15: 28.16 Asymptotic Expansions for Large $q$

… ►Let $s=2m+1$, $m=0,1,2,\mathrm{\dots }$, and $\nu$ be fixed with . … ►

28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

…

16: 26.10 Integer Partitions: Other Restrictions

… ►The set $\{2,3,4,\mathrm{\dots }\}$ is denoted by $T$. If more than one restriction applies, then the restrictions are separated by commas, for example, $p(𝒟2,\in T,n)$. … ►where the sum is over nonnegative integer values of $k$ for which $n-\frac{1}{2}(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $k$ for which $n-(3k^2\pm k)\ge 0$. … ►where the sum is over nonnegative integer values of $m$ for which $n-\frac{1}{2}k{m}^2-m+\frac{1}{2}km\ge 0$. …

17: 26.2 Basic Definitions

… ►Thus $231$ is the permutation $\sigma (1)=2$, $\sigma (2)=3$, $\sigma (3)=1$. … ►Here $\sigma (1)=2,\sigma (2)=5$, and $\sigma (5)=1$. … ►A lattice path is a directed path on the plane integer lattice $\{0,1,2,\mathrm{\dots }\}\times \{0,1,2,\mathrm{\dots }\}$. … ►As an example, $\{1,3,4\}$, $\{2,6\}$, $\{5\}$ is a partition of $\{1,2,3,4,5,6\}$. … ►As an example, $\{1,1,1,2,4,4\}$ is a partition of 13. …

18: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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5.17.3 $$G\left(z+1\right)={(2\pi )}^{z/2}\exp \left(-\frac{1}{2}z(z+1)-\frac{1}{2}\gamma z^2\right)\prod _{k=1}^{\mathrm{\infty }}\left({\left(1+\frac{z}{k}\right)}^k\exp \left(-z+\frac{z^2}{2k}\right)\right).$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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5.17.5 $$\mathrm{Ln}G\left(z+1\right)\sim \frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\ln z-\ln A+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant

Referenced by:

§5.17, §5.17, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5)

Permalink:

http://dlmf.nist.gov/5.17.E5

Encodings:

pMML, png, TeX

Errata (effective with 1.1.11):

For consistency we have replaced $\mathrm{Ln}z$ by $\ln z$.

Errata (effective with 1.0.7):

Originally the term $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.)

Reported 2013-08-01 by Gergő Nemes

See also:

Annotations for §5.17 and Ch.5

… ►Here $B_{2k+2}$ is the Bernoulli number (§24.2(i)), and $A$ is Glaisher’s constant, given by … ►

5.17.7 $$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$

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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, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant

Referenced by:

§5.17

Permalink:

http://dlmf.nist.gov/5.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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19: Bibliography G

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

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Mémoire sur les fonctions hypergéométriques d’ordre supérieur. Ann. Sci. École Norm. Sup. (2) 12, pp. 261–286, 395–430 (French).

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External Links:

ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

… ►

The solutions of Painlevé’s fifth equation. Differ. Uravn. 12 (4), pp. 740–742 (Russian).

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Notes:

In Russian. English translation: Differential Equations 12(4), pp. 519–521 (1977)

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(v).

Encodings:

BibTeX

►

One-parameter systems of solutions of Painlevé equations. Differ. Uravn. 14 (12), pp. 2131–2135 (Russian).

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Notes:

In Russian. English translation: Differential Equations 14(12), pp. 1510–1513 (1979)

External Links:

ISSN 0374-0641, MathReview (M. Tvrdý), ZentralBlatt

Referenced by:

§32.10(i), §32.10(ii), §32.10(iii), §32.7(iv).

Encodings:

BibTeX

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Algorithm 300: Coulomb wave functions. Comm. ACM 10 (4), pp. 244–245.

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Notes:

For remarks see same journal 12(1969), p. 692 and 16(1973), pp. 308-309 and for certification see same journal 12(1969), pp. 279-280.

External Links:

ISSN 0001-0782, Document

Referenced by:

§33.26(ii).

Encodings:

BibTeX

…

20: 5.23 Approximations

… ►Cody and Hillstrom (1967) gives minimax rational approximations for $\ln \mathrm{\Gamma }\left(x\right)$ for the ranges $0.5\le x\le 1.5$, $1.5\le x\le 4$, $4\le x\le 12$; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to $\mathrm{\Gamma }\left(x\right)$ and $\ln \mathrm{\Gamma }\left(x\right)$ in the intervals $0\le x\le 1$, $8\le x\le 1000$, $12\le x\le 1000$; precision is variable. … ►For additional approximations see Hart et al. (1968, Appendix B), Luke (1975, pp. 22–23), and Weniger (2003). … ►See Luke (1975, pp. 22–23) for additional expansions. …