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Gauss–Legendre formula

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1: 3.5 Quadrature
GaussLegendre Formula
Table 3.5.1: Nodes and weights for the 5-point GaussLegendre formula.
± x k w k
Table 3.5.2: Nodes and weights for the 10-point GaussLegendre formula.
± x k w k
Table 3.5.3: Nodes and weights for the 20-point GaussLegendre formula.
± x k w k
Table 3.5.5: Nodes and weights for the 80-point GaussLegendre formula.
± x k w k
2: 6.18 Methods of Computation
For an application of the Gauss-Legendre formula3.5(v)) see Tooper and Mark (1968). …
3: 18.5 Explicit Representations
§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 He n polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11). …
Legendre
4: Errata
  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Usability

    Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.

  • Subsections 15.4(i), 15.4(ii)

    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.

  • Subsection 14.18(iii)

    This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

  • Subsection 15.19(v)

    A new Subsection Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

  • 5: Frank W. J. Olver
    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. … , Bessel functions, hypergeometric functions, Legendre functions). … 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. …
    6: 15.9 Relations to Other Functions
    Legendre
    §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 R F and R G have connection formulas resulting from those for the Gauss hypergeometric function (Erdélyi et al. (1953a, §2.9)). … If 0 < p < z and y = z + 1 , then as p 0 (19.21.6) reduces to Legendre’s relation (19.21.1). … Connection formulas for R - a ( b ; z ) are given in Carlson (1977b, pp. 99, 101, and 123–124). …
    8: Bibliography G
  • F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
  • C. F. Gauss (1863) Werke. Band II. pp. 436–447 (German).
  • W. Gautschi (2002b) Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions. J. Comput. Appl. Math. 139 (1), pp. 173–187.
  • G. H. Golub and J. H. Welsch (1969) Calculation of Gauss quadrature rules. Math. Comp. 23 (106), pp. 221–230.
  • H. W. Gould (1972) Explicit formulas for Bernoulli numbers. Amer. Math. Monthly 79, pp. 44–51.
  • 9: Bibliography V
  • A. J. van der Poorten (1980) 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.
  • A. Verma and V. K. Jain (1983) Certain summation formulae for q -series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
  • R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • N. Virchenko and I. Fedotova (2001) Generalized Associated Legendre Functions and their Applications. World Scientific Publishing Co. Inc., Singapore.
  • 10: 18.12 Generating Functions
    and a similar formula by symmetry; compare the second row in Table 18.6.1. For the hypergeometric function F 1 2 see §§15.1, 15.2(i). …
    Legendre
    18.12.11 1 1 - 2 x z + z 2 = n = 0 P n ( x ) z n , | z | < 1 .
    18.12.12 e x z J 0 ( z 1 - x 2 ) = n = 0 P n ( x ) n ! z n .