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1: 18.42 Software
For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3). …
2: 7.24 Approximations
§7.24 Approximations
§7.24(i) Approximations in Terms of Elementary Functions
  • 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).

  • 3: 25.20 Approximations
    §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.

  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • 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 .

  • 4: 6.20 Approximations
    §6.20 Approximations
    §6.20(i) Approximations in Terms of Elementary Functions
  • 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.

  • 5: 16.26 Approximations
    §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).
    6: 31.13 Asymptotic Approximations
    §31.13 Asymptotic Approximations
    For asymptotic approximations for the accessory parameter eigenvalues q m , see Fedoryuk (1991) and Slavyanov (1996). For asymptotic approximations of the solutions of Heun’s equation (31.2.1) when two singularities are close together, see Lay and Slavyanov (1999). For asymptotic approximations of the solutions of confluent forms of Heun’s equation in the neighborhood of irregular singularities, see Komarov et al. (1976), Ronveaux (1995, Parts B,C,D,E), Bogush and Otchik (1997), Slavyanov and Veshev (1997), and Lay et al. (1998).
    7: 10.76 Approximations
    §10.76 Approximations
    §10.76(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions
    Bickley Functions
    Spherical Bessel Functions
    Kelvin Functions
    8: 20 Theta Functions
    Chapter 20 Theta Functions
    9: 4.47 Approximations
    §4.47 Approximations
    §4.47(iii) Padé Approximations
    Luke (1975, Chapter 3) supplies real and complex approximations for ln , exp , sin , cos , tan , arctan , arcsinh . …
    10: 9.13 Generalized Airy Functions
    are used in approximating solutions to differential equations with multiple turning points; see §2.8(v). … As z Reid (1972) and Drazin and Reid (1981, Appendix) introduce the following contour integrals in constructing approximate solutions to the Orr–Sommerfeld equation for fluid flow: …
    9.13.32 f ( p 3 ) z f ( p 1 ) + ( p 1 ) f ( p ) = 0 .