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1: 7.24 Approximations
  • 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).

  • 2: Bibliography N
  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
  • 3: Bibliography D
  • B. Döring (1966) Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.
  • T. M. Dunster (1989) Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.
  • 4: 7.23 Tables
  • Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of erf z , x [ 0 , 5 ] , y = 0.5 ( .5 ) 3 , 7D and 8D, respectively; the real and imaginary parts of x e ± i t 2 d t , ( 1 / π ) e i ( x 2 + ( π / 4 ) ) x e ± i t 2 d t , x = 0 ( .5 ) 20 ( 1 ) 25 , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

  • 5: 5.11 Asymptotic Expansions
    Wrench (1968) gives exact values of g k up to g 20 . …
    §5.11(ii) Error Bounds and Exponential Improvement
    For error bounds for (5.11.8) and an exponentially-improved extension, see Nemes (2013b). … For further information see Olver (1997b, pp. 293–295), and for other error bounds see Whittaker and Watson (1927, §12.33), Spira (1971), and Schäfke and Finsterer (1990). … For realistic error bounds in (5.11.14) see Frenzen (1987a, 1992). …
    6: Bibliography F
  • FDLIBM (free C library)
  • H. E. Fettis, J. C. Caslin, and K. R. Cramer (1973) Complex zeros of the error function and of the complementary error function. Math. Comp. 27 (122), pp. 401–407.
  • C. L. Frenzen (1987a) Error bounds for asymptotic expansions of the ratio of two gamma functions. SIAM J. Math. Anal. 18 (3), pp. 890–896.
  • C. L. Frenzen (1990) Error bounds for a uniform asymptotic expansion of the Legendre function Q n m ( cosh z ) . SIAM J. Math. Anal. 21 (2), pp. 523–535.
  • C. L. Frenzen (1992) Error bounds for the asymptotic expansion of the ratio of two gamma functions with complex argument. SIAM J. Math. Anal. 23 (2), pp. 505–511.
  • 7: Bibliography S
  • K. L. Sala (1989) Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.
  • J. L. Schonfelder (1978) Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.
  • A. Sharples (1967) Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
  • J. R. Stembridge (1995) A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.
  • A. Strecok (1968) On the calculation of the inverse of the error function. Math. Comp. 22 (101), pp. 144–158.
  • 8: Bibliography O
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.
  • F. W. J. Olver (1964b) Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.
  • F. W. J. Olver (1974) Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.
  • F. W. J. Olver (1980a) Asymptotic approximations and error bounds. SIAM Rev. 22 (2), pp. 188–203.
  • 9: Bibliography V
  • C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.
  • R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.
  • R. Vidūnas (2005) Transformations of some Gauss hypergeometric functions. J. Comput. Appl. Math. 178 (1-2), pp. 473–487.
  • N. Ja. Vilenkin (1968) Special Functions and the Theory of Group Representations. American Mathematical Society, Providence, RI.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • 10: 30.9 Asymptotic Approximations and Expansions
    §30.9(i) Prolate Spheroidal Wave Functions
    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