Gauss formula

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1: 3.5 Quadrature

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Gauss–Legendre Formula

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Gauss–Chebyshev Formula

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Gauss–Jacobi Formula

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Gauss–Laguerre Formula

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Gauss–Hermite Formula

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2: 6.18 Methods of Computation

… ►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968). …

3: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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Gauss Formula

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4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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Pfaff–Saalschütz Formula

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5: 5.5 Functional Relations

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Gauss’s Multiplication Formula

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6: 15.4 Special Cases

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15.4.1 $$F(1,1;2;z)=-z^{-1}\ln \left(1-z\right),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Referenced by:

§15.4(i)

Permalink:

http://dlmf.nist.gov/15.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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15.4.2 $$F(\frac{1}{2},1;\frac{3}{2};z^2)=\frac{1}{2z}\ln \left(\frac{1+z}{1-z}\right),$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\ln z$: principal branch of logarithm function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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15.4.3 $$F(\frac{1}{2},1;\frac{3}{2};-z^2)=z^{-1}\arctan z,$$

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Symbols:

$F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\arctan z$: arctangent function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/15.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.4(i), §15.4 and Ch.15

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$$F(a,b;a;z)={(1-z)}^{-b},$$

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$$F(a,b;b;z)={(1-z)}^{-a},$$

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7: 16.8 Differential Equations

… ►Analytical continuation formulas for ${}_{q+1}{}^{}F_q^{}(𝐚;𝐛;z)$ near $z=1$ are given in Bühring (1987b) for the case $q=2$, and in Bühring (1992) for the general case. …

8: 15.2 Definitions and Analytical Properties

… ►Formula (15.4.6) reads $F(a,b;a;z)={(1-z)}^{-b}$. …

9: Bibliography G

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Construction of Gauss-Christoffel quadrature formulas. Math. Comp. 22, pp. 251–270.

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External Links:

ISSN 0025-5718, Document, Link, MathReview (R. E. Barnhill)

Referenced by:

§18.39(iii).

Encodings:

BibTeX

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10: Errata

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Sentences were added specifying that some equations in these subsections require special care under certain circumstances. Also, (15.4.6) was expanded by adding the formula $F(a,b;a;z)={(1-z)}^{-b}$.

Report by Louis Klauder on 2017-01-01.

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