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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
  • Other Changes


    • Equations (4.45.8) and (4.45.9) have been replaced with equations that are better for numerically computing arctan x .

    • A new Subsection 13.29(v) Continued Fractions, has been added to cover computation of confluent hypergeometric functions by continued fractions.

    • A new Subsection 14.5(vi) Addendum to §14.5(ii) μ = 0 , ν = 2 , containing the values of Legendre and Ferrers functions for degree ν = 2 has been added.

    • Subsection 14.18(iii) has been altered to identify Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

    • A new Subsection 15.19(v) Continued Fractions, has been added to cover computation of the Gauss hypergeometric functions by continued fractions.

    • Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

    • Cross-references have been added in §§1.2(i), 10.19(iii), 10.23(ii), 17.2(iii), 18.15(iii), 19.2(iv), 19.16(i).

    • Several small revisions have been made. For details see §§5.11(ii), 10.12, 10.19(ii), 18.9(i), 18.16(iv), 19.7(ii), 22.2, 32.11(v), 32.13(ii).

    • Entries for the Sage computational system have been updated in the Software Index.

    • The default document format for DLMF is now HTML5 which includes MathML providing better accessibility and display of mathematics.

    • All interactive 3D graphics on the DLMF website have been recast using WebGL and X3DOM, improving portability and performance; WebGL it is now the default format.

  • 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: 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 .
    10: Bibliography Z
  • A. Zarzo, J. S. Dehesa, and R. J. Yañez (1995) Distribution of zeros of Gauss and Kummer hypergeometric functions. A semiclassical approach. Ann. Numer. Math. 2 (1-4), pp. 457–472.
  • M. I. Žurina and L. N. Karmazina (1963) Tablitsy funktsii Lezhandra P - 1 / 2 + i τ 1 ( x ) . Vyčisl. Centr Akad. Nauk SSSR, Moscow.
  • M. I. Žurina and L. N. Karmazina (1964) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.
  • M. I. Žurina and L. N. Karmazina (1965) Tables of the Legendre functions P - 1 / 2 + i τ ( x ) . Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.
  • M. I. Žurina and L. N. Karmazina (1966) Tables and formulae for the spherical functions P - 1 / 2 + i τ m ( z ) . Translated by E. L. Albasiny, Pergamon Press, Oxford.