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7 Error Functions, Dawson’s and Fresnel IntegralsComputation

§7.24 Approximations


§7.24(i) Approximations in Terms of Elementary Functions

  • Hastings (1955) gives several minimax polynomial and rational approximations for erfx, erfcx and the auxiliary functions f(x) and g(x).

  • Cody (1969) provides minimax rational approximations for erfx and erfcx. 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).

§7.24(ii) Expansions in Chebyshev Series

  • Luke (1969b, pp. 323–324) covers 12πerfx and ex2F(x) for -3x3 (the Chebyshev coefficients are given to 20D); πxex2erfcx and 2xF(x) for x3 (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

  • Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions f(x) and g(x) for x3 (15D).

  • Schonfelder (1978) gives coefficients of Chebyshev expansions for x-1erfx on 0x2, for xex2erfcx on [2,), and for ex2erfcx on [0,) (30D).

  • Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for (1+2x)ex2erfcx on (0,) (22D).

§7.24(iii) Padé-Type Expansions

  • Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for F(z), erfz, erfcz, C(z), and S(z); approximate errors are given for a selection of z-values.