§7.24 Approximations

§7.24(i) Approximations in Terms of Elementary Functions

Keywords:: Dawson’s integral, Fresnel integrals, auxiliary functions for Fresnel integrals, error functions
Permalink:: http://dlmf.nist.gov/7.24.i

Hastings (1955) gives several minimax polynomial and rational approximations for $\mathop{\mathrm{erf}\/}\nolimits x$ , $\mathop{\mathrm{erfc}\/}\nolimits x$ and the auxiliary functions $\mathop{\mathrm{f}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{g}\/}\nolimits\!\left(x\right)$ .
Cody (1969) provides minimax rational approximations for $\mathop{\mathrm{erf}\/}\nolimits x$ and $\mathop{\mathrm{erfc}\/}\nolimits x$ . 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 $\mathop{F\/}\nolimits\!\left(x\right)$ (maximum relative precision 20S–22S).

Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi}\mathop{\mathrm{erf}\/}\nolimits x$ and $e^{{x^{2}}}\mathop{F\/}\nolimits\!\left(x\right)$ for $-3\leq x\leq 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi}xe^{{x^{2}}}\mathop{\mathrm{erfc}\/}\nolimits x$ and $2x\mathop{F\/}\nolimits\!\left(x\right)$ for $x\geq 3$ (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 $\mathop{\mathrm{f}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{g}\/}\nolimits\!\left(x\right)$ for $x\geq 3$ (15D).
Schonfelder (1978) gives coefficients of Chebyshev expansions for $x^{{-1}}\mathop{\mathrm{erf}\/}\nolimits x$ on $0\leq x\leq 2$ , for $xe^{{x^{2}}}\mathop{\mathrm{erfc}\/}\nolimits x$ on $[2,\infty)$ , and for $e^{{x^{2}}}\mathop{\mathrm{erfc}\/}\nolimits x$ on $[0,\infty)$ (30D).
Shepherd and Laframboise (1981) gives coefficients of Chebyshev series for $(1+2x)e^{{x^{2}}}\mathop{\mathrm{erfc}\/}\nolimits x$ on $(0,\infty)$ (22D).

Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $\mathop{F\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{erf}\/}\nolimits z$ , $\mathop{\mathrm{erfc}\/}\nolimits z$ , $\mathop{C\/}\nolimits\!\left(z\right)$ , and $\mathop{S\/}\nolimits\!\left(z\right)$ ; approximate errors are given for a selection of -values.