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1: 5.23 Approximations
§5.23(i) Rational Approximations
Cody et al. (1973) gives minimax rational approximations for ψ ( x ) for the ranges 0.5 x 3 and 3 x < ; precision is variable. …
§5.23(iii) Approximations in the Complex Plane
See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of Γ ( z ) . For rational approximations to ψ ( z ) + γ see Luke (1975, pp. 13–16).
2: 16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer G -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
3: 16.7 Relations to Other Functions
Further representations of special functions in terms of F q p functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of F q q + 1 functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
4: 6.20 Approximations
  • 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.

  • §6.20(iii) Padé-Type and Rational Expansions
  • Luke (1969b, pp. 411–414) gives rational approximations for Ein ( z ) .

  • 5: 5.19 Mathematical Applications
    §5.19(i) Summation of Rational Functions
    As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions. …
    6: 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.

  • Morris (1979) gives rational approximations for Li 2 ( x ) 25.12(i)) for 0.5 x 1 . Precision is varied with a maximum of 24S.

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals < x 2 and 2 x < , with s = 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 4 , 10 8 , 10 12 .

  • 7: 4.47 Approximations
    §4.47(ii) Rational Functions
    8: 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 (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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

  • 9: 33.25 Approximations
    Cody and Hillstrom (1970) provides rational approximations of the phase shift σ 0 ( η ) = ph Γ ( 1 + i η ) (see (33.2.10)) for the ranges 0 η 2 , 2 η 4 , and 4 η . …
    10: Annie A. M. Cuyt
    A lot of her research has been devoted to rational approximations, in one as well as in many variables, and sparse interpolation. …