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21: 6.20 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for E 1 ( x ) + ln x , x e x E 1 ( x ) , and the auxiliary functions f ( x ) and g ( x ) . These are included in Abramowitz and Stegun (1964, Ch. 5).

  • Cody and Thacher (1968) provides minimax rational approximations for E 1 ( x ) , with accuracies up to 20S.

  • Cody and Thacher (1969) provides minimax rational approximations for Ei ( x ) , with accuracies up to 20S.

  • MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions f and g , with accuracies up to 20S.

  • 22: 24.18 Physical Applications
    §24.18 Physical Applications
    Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
    23: 7.24 Approximations
  • Hastings (1955) gives several minimax polynomial and rational approximations for erf x , erfc x and the auxiliary functions f ( x ) and g ( x ) .

  • Cody (1969) provides minimax rational approximations for erf x and erfc x . The maximum relative precision is about 20S.

  • Cody et al. (1970) gives minimax rational approximations to Dawson’s integral F ( x ) (maximum relative precision 20S–22S).

  • 24: 32.8 Rational Solutions
    where the Q n ( z ) are monic polynomials (coefficient of highest power of z is 1 ) satisfying … Next, let p m ( z ) be the polynomials defined by p m ( z ) = 0 for m < 0 , and … where P m ( z ) and Q m ( z ) are polynomials of degree m , with no common zeros. … where P j , n 1 ( z ) and Q j , n ( z ) are polynomials of degrees n 1 and n , respectively, with no common zeros. … where λ , μ are constants, and P n 1 ( z ) , Q n ( z ) are polynomials of degrees n 1 and n , respectively, with no common zeros. …
    25: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • 26: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • C. Krattenthaler (1993) HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively q -binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.
  • 27: 24.3 Graphs
    See accompanying text
    Figure 24.3.1: Bernoulli polynomials B n ( x ) , n = 2 , 3 , , 6 . Magnify
    See accompanying text
    Figure 24.3.2: Euler polynomials E n ( x ) , n = 2 , 3 , , 6 . Magnify
    28: 18.4 Graphics
    See accompanying text
    Figure 18.4.1: Jacobi polynomials P n ( 1.5 , 0.5 ) ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
    See accompanying text
    Figure 18.4.2: Jacobi polynomials P n ( 1.25 , 0.75 ) ( x ) , n = 7 , 8 . … Magnify
    See accompanying text
    Figure 18.4.4: Legendre polynomials P n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
    See accompanying text
    Figure 18.4.5: Laguerre polynomials L n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
    See accompanying text
    Figure 18.4.7: Monic Hermite polynomials h n ( x ) = 2 n H n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
    29: 18.7 Interrelations and Limit Relations
    §18.7 Interrelations and Limit Relations
    Chebyshev, Ultraspherical, and Jacobi
    Legendre, Ultraspherical, and Jacobi
    §18.7(ii) Quadratic Transformations
    §18.7(iii) Limit Relations
    30: 18.41 Tables
    §18.41(i) Polynomials
    For P n ( x ) ( = 𝖯 n ( x ) ) see §14.33. Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates T n ( x ) , U n ( x ) , L n ( x ) , and H n ( x ) for n = 0 ( 1 ) 12 . The ranges of x are 0.2 ( .2 ) 1 for T n ( x ) and U n ( x ) , and 0.5 , 1 , 3 , 5 , 10 for L n ( x ) and H n ( x ) . … For P n ( x ) , L n ( x ) , and H n ( x ) see §3.5(v). …