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31: 16.13 Appell Functions

§16.13 Appell Functions

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32: 25.10 Zeros

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25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

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25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

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See also:

Annotations for §25.10(i), §25.10 and Ch.25

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33: 26.7 Set Partitions: Bell Numbers

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§26.7(i) Definitions

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34: 8.2 Definitions and Basic Properties

§8.2 Definitions and Basic Properties

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§8.2(i) Definitions

►The general values of the incomplete gamma functions $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ are defined by …However, when the integration paths do not cross the negative real axis, and in the case of (8.2.2) exclude the origin, $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ take their principal values; compare §4.2(i). …

35: 18.19 Hahn Class: Definitions

§18.19 Hahn Class: Definitions

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Hahn class (or linear lattice class). These are OP’s $p_n(x)$ where the role of $\frac{d}{dx}$ is played by ${\mathrm{\Delta }}_x$ or ${\nabla }_x$ or ${\delta }_x$ (see §18.1(i) for the definition of these operators). The Hahn class consists of four discrete and two continuous families.

… ►Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and standardization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_n(x;\alpha ,\beta ,N)$, Krawtchouk polynomials $K_n(x;p,N)$, Meixner polynomials $M_n(x;\beta ,c)$, and Charlier polynomials $C_n(x;a)$. … ►A special case of (18.19.8) is $w^{(1/2)}(x;\pi /2)=\frac{\pi }{\cosh \left(\pi x\right)}$.

36: 25.2 Definition and Expansions

§25.2 Definition and Expansions

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§25.2(i) Definition

►When $\mathrm{\Re }s>1$, ►

25.2.1 $$\zeta \left(s\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}.$$

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Defines:

$\zeta \left(s\right)$: Riemann zeta function

Symbols:

$n$: nonnegative integer and $s$: complex variable

Keywords:

definition, infinite series

Sources:

Apostol (1976, Chapter 12); Riemann (1859, p. 136)

A&S Ref:

23.2.1

Referenced by:

(25.11.2), (25.2.2), (25.2.4), §25.2(ii), Erratum (V1.0.17) for Equation (25.2.4)

Permalink:

http://dlmf.nist.gov/25.2.E1

Encodings:

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See also:

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

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25.2.5 $${\gamma }_n=\underset{m\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^m\frac{{(\ln k)}^n}{k}-\frac{{(\ln m)}^{n+1}}{n+1}\right).$$

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Defines:

${\gamma }_n$: Stieltjes constants

Symbols:

$\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer

Keywords:

definition, Stieltjes constants

Sources:

Hardy (1912, p. 215); Ivić (1985, (1.12), p. 4)

Referenced by:

§25.2(ii), §25.6(ii), Erratum (V1.0.23) for Subsection 25.2(ii) Other Infinite Series

Permalink:

http://dlmf.nist.gov/25.2.E5

Encodings:

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See also:

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

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37: 25.9 Asymptotic Approximations

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25.9.2 $$\chi (s)\equiv {\pi }^{s-\frac{1}{2}}\mathrm{\Gamma }\left(\frac{1}{2}-\frac{1}{2}s\right)/\mathrm{\Gamma }\left(\frac{1}{2}s\right).$$

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Symbols:

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

Keywords:

definition

Source:

Titchmarsh (1986b, p. 78); with (5.5.5), (5.5.3)

Referenced by:

(25.10.2)

Permalink:

http://dlmf.nist.gov/25.9.E2

Encodings:

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See also:

Annotations for §25.9 and Ch.25

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38: 25.13 Periodic Zeta Function

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25.13.1 $$F(x,s)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}nx}}{n^s},$$

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Defines:

$F(x,s)$: periodic zeta function

Symbols:

$\pi$: 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, $x$: real variable and $s$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, (9), p. 257)

Permalink:

http://dlmf.nist.gov/25.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

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39: 30.5 Functions of the Second Kind

§30.5 Functions of the Second Kind

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40: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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

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§24.2(ii) Euler Numbers and Polynomials

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

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