n-tuple formulas

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11: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the

icon) for links to defining formula. …

12: 18.42 Software

… ►A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …

13: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

14: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

15: 3.5 Quadrature

… ► … ►

Gauss–Legendre Formula

… ►

Gauss–Chebyshev Formula

… ►

Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. …

16: 25.4 Reflection Formulas

§25.4 Reflection Formulas

… ►

25.4.1

\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac{1}{2}\pi s\right)\Gamma% \left(s\right)\zeta\left(s\right),

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (12), p. 259)
Referenced by:: §25.10(i), §5.17
Permalink:: http://dlmf.nist.gov/25.4.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

►

25.4.2

\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{1}{2}\pi s\right)\Gamma\left% (1-s\right)\zeta\left(1-s\right).

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\sin\NVar{z}$ : sine function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, (13), p. 259)
A&S Ref:: 23.2.6
Permalink:: http://dlmf.nist.gov/25.4.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

… ►

25.4.3

\xi\left(s\right)=\xi\left(1-s\right),

ⓘ

Symbols:: $\xi\left(\NVar{s}\right)$ : Riemann’s $\xi$ -function and $s$ : complex variable
Keywords:: reflection formula
Source:: Apostol (1976, p. 260)
Permalink:: http://dlmf.nist.gov/25.4.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §25.4 and Ch.25

…

17: 2.2 Transcendental Equations

… ►where

F_{0}=f_{0}

and

sF_{s}

(

s\geq 1

) is the coefficient of

x^{-1}

in the asymptotic expansion of

(f(x))^{s}

(Lagrange’s formula for the reversion of series). …

18: 25.19 Tables

… ►

•

Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ , $x=-5(.05)25$ , to 12S.

►

•

Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$ . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$ , and §22.17 lists tables for some Dirichlet $L$ -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

19: 29.20 Methods of Computation

… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►§29.15(i) includes formulas for normalizing the eigenvectors. …

20: 14.28 Sums

… ►

§14.28(ii) Heine’s Formula

…