Gauss–Legendre formula
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1: 3.5 Quadrature
2: 6.18 Methods of Computation
3: 18.5 Explicit Representations
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§18.5(ii) Rodrigues Formulas
… ►Related formula: … ►and two similar formulas by symmetry; compare the second row in Table 18.6.1. … ►For corresponding formulas for Chebyshev, Legendre, and the Hermite polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). … ►Legendre
…4: Errata
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Paragraph Inversion Formula (in §35.2)
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Usability
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Subsections 15.4(i), 15.4(ii)
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Subsection 14.18(iii)
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Subsection 15.19(v)
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The wording was changed to make the integration variable more apparent.
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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 .
Report by Louis Klauder on 2017-01-01.
A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.
5: Frank W. J. Olver
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►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history.
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►, Bessel functions, hypergeometric functions, Legendre functions).
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►In a review of that volume, Jet Wimp of Drexel University said that the papers “exemplify a redoubtable mathematical talent, the work of a man who has done more than almost anyone else in the 20th century to bestow on the discipline of applied mathematics the elegance and rigor that its earliest practitioners, such as Gauss and Laplace, would have wished for it.
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6: 15.9 Relations to Other Functions
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Legendre
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15.9.7
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§15.9(iv) Associated Legendre Functions; Ferrers Functions
►Any hypergeometric function for which a quadratic transformation exists can be expressed in terms of associated Legendre functions or Ferrers functions. … ►The following formulas apply with principal branches of the hypergeometric functions, associated Legendre functions, and fractional powers. …7: 19.21 Connection Formulas
§19.21 Connection Formulas
… ►Legendre’s relation (19.7.1) can be written … ►The complete cases of and have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … ►If and , then as (19.21.6) reduces to Legendre’s relation (19.21.1). … ►Connection formulas for are given in Carlson (1977b, pp. 99, 101, and 123–124). …8: Bibliography G
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Contiguous relations and summation and transformation formulae for basic hypergeometric series.
J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
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Werke. Band II.
pp. 436–447 (German).
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Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions.
J. Comput. Appl. Math. 139 (1), pp. 173–187.
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Calculation of Gauss quadrature rules.
Math. Comp. 23 (106), pp. 221–230.
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Explicit formulas for Bernoulli numbers.
Amer. Math. Monthly 79, pp. 44–51.
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9: Bibliography V
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Some Wonderful Formulas
an Introduction to Polylogarithms.
In Proceedings of the Queen’s Number Theory Conference, 1979
(Kingston, Ont., 1979), R. Ribenboim (Ed.),
Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
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Certain summation formulae for -series.
J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
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Transformations of some Gauss hypergeometric functions.
J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
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Generalized Associated Legendre Functions and their Applications.
World Scientific Publishing Co. Inc., Singapore.
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