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11: 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 .

  • 12: 7.24 Approximations
    §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).

  • 13: 10.57 Uniform Asymptotic Expansions for Large Order
    §10.57 Uniform Asymptotic Expansions for Large Order
    Asymptotic expansions for 𝗃 n ( ( n + 1 2 ) z ) , 𝗒 n ( ( n + 1 2 ) z ) , 𝗁 n ( 1 ) ( ( n + 1 2 ) z ) , 𝗁 n ( 2 ) ( ( n + 1 2 ) z ) , 𝗂 n ( 1 ) ( ( n + 1 2 ) z ) , and 𝗄 n ( ( n + 1 2 ) z ) as n that are uniform with respect to z can be obtained from the results given in §§10.20 and 10.41 by use of the definitions (10.47.3)–(10.47.7) and (10.47.9). …
    14: 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 ) .

  • 15: Bibliography L
  • C. Leubner and H. Ritsch (1986) A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.
  • J. L. López (2001) Uniform asymptotic expansions of symmetric elliptic integrals. Constr. Approx. 17 (4), pp. 535–559.
  • J. L. López and N. M. Temme (1999c) Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.
  • D. Ludwig (1966) Uniform asymptotic expansions at a caustic. Comm. Pure Appl. Math. 19, pp. 215–250.
  • Y. L. Luke (1977a) Algorithms for rational approximations for a confluent hypergeometric function. Utilitas Math. 11, pp. 123–151.
  • 16: 10.72 Mathematical Applications
    Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. … These expansions are uniform with respect to z , including the turning point z 0 and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … If f ( z ) has a double zero z 0 , or more generally z 0 is a zero of order m , m = 2 , 3 , 4 , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order 1 / ( m + 2 ) . … These asymptotic expansions are uniform with respect to z , including cut neighborhoods of z 0 , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. … These approximations are uniform with respect to both z and α , including z = z 0 ( a ) , the cut neighborhood of z = 0 , and α = a . …
    17: Bibliography R
  • A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
  • M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.
  • W. H. Reid (1972) Composite approximations to the solutions of the Orr-Sommerfeld equation. Studies in Appl. Math. 51, pp. 341–368.
  • W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
  • W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.
  • 18: 34.8 Approximations for Large Parameters
    §34.8 Approximations for Large Parameters
    Semiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). Uniform approximations in terms of Airy functions for the 3 j and 6 j symbols are given in Schulten and Gordon (1975b). For approximations for the 3 j , 6 j , and 9 j symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.
    19: Bibliography B
  • A. P. Bassom, P. A. Clarkson, C. K. Law, and J. B. McLeod (1998) Application of uniform asymptotics to the second Painlevé transcendent. Arch. Rational Mech. Anal. 143 (3), pp. 241–271.
  • M. V. Berry (1966) Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.
  • M. V. Berry (1969) Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.
  • J. M. Blair, C. A. Edwards, and J. H. Johnson (1976) Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.
  • J. M. Blair, C. A. Edwards, and J. H. Johnson (1978) Rational Chebyshev approximations for the Bickley functions K i n ( x ) . Math. Comp. 32 (143), pp. 876–886.
  • 20: 10.69 Uniform Asymptotic Expansions for Large Order
    §10.69 Uniform Asymptotic Expansions for Large Order
    All fractional powers take their principal values. …