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1: 25.5 Integral Representations
§25.5 Integral Representations
§25.5(i) In Terms of Elementary Functions
§25.5(ii) In Terms of Other Functions
For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). In (25.5.15)–(25.5.19), 0 < s < 1 , ψ ( x ) is the digamma function, and γ is Euler’s constant (§5.2). …
2: 7.24 Approximations
§7.24(i) Approximations in Terms of Elementary Functions
  • 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).

  • §7.24(ii) Expansions in Chebyshev Series
    3: 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.

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • 4: 19.36 Methods of Computation
    When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …where the elementary symmetric functions E s are defined by (19.19.4). If (19.36.1) is used instead of its first five terms, then the factor ( 3 r ) 1 / 6 in Carlson (1995, (2.2)) is changed to ( 3 r ) 1 / 8 . … If α 2 = k 2 , then the method fails, but the function can be expressed by (19.6.13) in terms of E ( ϕ , k ) , for which Neuman (1969b) uses ascending Landen transformations. … Numerical quadrature is slower than most methods for the standard integrals but can be useful for elliptic integrals that have complicated representations in terms of standard integrals. …
    5: 12.10 Uniform Asymptotic Expansions for Large Parameter
    With the upper sign in (12.10.2), expansions can be constructed for large μ in terms of elementary functions that are uniform for t ( , ) 2.8(ii)). … The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions2.8(iii)). …
    §12.10(ii) Negative a , 2 a < x <
    §12.10(vi) Modifications of Expansions in Elementary Functions
    Modified Expansions
    6: Bibliography G
  • W. Gautschi (1959b) Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.
  • W. Gautschi (1967) Computational aspects of three-term recurrence relations. SIAM Rev. 9 (1), pp. 24–82.
  • W. Gautschi (1975) Computational Methods in Special Functions – A Survey. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), R. A. Askey (Ed.), pp. 1–98. Math. Res. Center, Univ. Wisconsin Publ., No. 35.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • GSL (free C library) GNU Scientific Library The GNU Project.
  • 7: Bibliography C
  • A. Cayley (1895) An Elementary Treatise on Elliptic Functions. George Bell and Sons, London.
  • A. Cayley (1961) An Elementary Treatise on Elliptic Functions. Dover Publications, New York (English).
  • T. M. Cherry (1948) Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.
  • W. J. Cody and W. Waite (1980) Software Manual for the Elementary Functions. Prentice-Hall, Englewood Cliffs.
  • W. J. Cody (1993a) Algorithm 714: CELEFUNT – A portable test package for complex elementary functions. ACM Trans. Math. Software 19 (1), pp. 1–21.
  • 8: 30.9 Asymptotic Approximations and Expansions
    For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). …
    §30.9(ii) Oblate Spheroidal Wave Functions
    For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). …
    §30.9(iii) Other Approximations and Expansions
    The asymptotic behavior of λ n m ( γ 2 ) and a n , k m ( γ 2 ) as n in descending powers of 2 n + 1 is derived in Meixner (1944). …
    9: Bibliography M
  • T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.
  • S. M. Markov (1981) On the interval computation of elementary functions. C. R. Acad. Bulgare Sci. 34 (3), pp. 319–322.
  • X. Merrheim (1994) The computation of elementary functions in radix 2 p . Computing 53 (3-4), pp. 219–232.
  • H. J. W. Müller (1966b) Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.
  • J. Muller (1997) Elementary Functions: Algorithms and Implementation. Birkhäuser Boston Inc., Boston, MA.