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31: Mathematical Introduction

… ►See, for example, Chapters 16, 17, 18, 19, 21, 27, 29, 31, 32, 34, 35, and 36. …

32: 24.16 Generalizations

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33: 25.2 Definition and Expansions

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25.2.12 $$\zeta \left(s\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{-s-(\gamma s/2)}}{2(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s+1\right)}\prod _{\rho }\left(1-\frac{s}{\rho }\right){\mathrm{e}}^{s/\rho },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $s$: complex variable and $\rho$: zeros

Keywords:

infinite product, primes

Source:

Titchmarsh (1986b, (2.12.6), (2.12.8), p. 30–31)

A&S Ref:

23.2.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

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34: 26.10 Integer Partitions: Other Restrictions

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

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	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
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$20$	$64$	$31$	$20$	$18$

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35: Bibliography K

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Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.

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External Links:

ISSN 0305-4470, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

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On the zeroes of the Pearcey integral. J. Comput. Appl. Math. 107 (1), pp. 31–52.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§36.7(ii), §36.7(ii).

Encodings:

BibTeX

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36: 1.12 Continued Fractions

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37: 18.38 Mathematical Applications

… ►For these results and applications in approximation theory see §3.11(ii) and Mason and Handscomb (2003, Chapter 3), Cheney (1982, p. 108), and Rivlin (1969, p. 31). …

38: 24.2 Definitions and Generating Functions

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Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

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	$k$
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$9$	$-\frac{31}{2}$	$0$	$\frac{153}{2}$	$0$	$-63$	$0$	$21$	$0$	$-\frac{9}{2}$	$1$
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39: 25.12 Polylogarithms

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25.12.13 $${\mathrm{Li}}_s\left({\mathrm{e}}^{2\pi \mathrm{i}a}\right)+{\mathrm{e}}^{\pi \mathrm{i}s}{\mathrm{Li}}_s\left({\mathrm{e}}^{-2\pi \mathrm{i}a}\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{\pi \mathrm{i}s/2}}{\mathrm{\Gamma }\left(s\right)}\zeta (1-s,a),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $(a,b)$: open interval, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable

Source:

Erdélyi et al. (1953a, (1.11.16), p. 31); with ${\mathrm{Li}}_s\left(z\right)=F(z,s)$, $z={\mathrm{e}}^{2\pi \mathrm{i}a}$

Referenced by:

§25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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40: 26.12 Plane Partitions

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Table 26.12.1: Plane partitions.

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$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$	$n$	$\mathrm{pp}\left(n\right)$
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14	4167	31	85 12309	48	51913 04973
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