Leibniz formula

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11—20 of 243 matching pages

11: 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. …

12: 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. …

13: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Chebyshev Formula

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Gauss–Laguerre Formula

… ►a complex Gauss quadrature formula is available. …

14: 25.4 Reflection Formulas

§25.4 Reflection Formulas

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25.4.1 $$\zeta \left(1-s\right)=2{(2\pi )}^{-s}\cos \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(s\right)\zeta \left(s\right),$$

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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, $\cos z$: 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

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25.4.2 $$\zeta \left(s\right)=2{(2\pi )}^{s-1}\sin \left(\frac{1}{2}\pi s\right)\mathrm{\Gamma }\left(1-s\right)\zeta \left(1-s\right).$$

ⓘ

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, $\sin z$: 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

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25.4.3 $$\xi \left(s\right)=\xi \left(1-s\right),$$

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Symbols:

$\xi \left(s\right)$: 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

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15: 2.2 Transcendental Equations

… ►where $F_0=f_0$ and $s{F}_s$ ($s\ge 1$) is the coefficient of $x^{-1}$ in the asymptotic expansion of ${(f(x))}^s$ (Lagrange’s formula for the reversion of series). …

16: 25.19 Tables

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, 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.

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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 $\zeta (s,a)$, 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.

17: 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. …

18: 14.28 Sums

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§14.28(ii) Heine’s Formula

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19: 17.2 Calculus

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Leibniz Rule

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20: Need Help?

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