error%20functions
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11: 25.20 Approximations
Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).
12: 9.18 Tables
Miller (1946) tabulates , for , for ; , for ; , for ; , , , (respectively , , , ) for . Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.
Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
Sherry (1959) tabulates , , , , ; 20S.
§9.18(vi) Scorer Functions
…13: 7.8 Inequalities
14: 11.6 Asymptotic Expansions
§11.6(i) Large , Fixed
… ►§11.6(ii) Large , Fixed
… ► … ►Here … ►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).15: Bibliography G
16: Bibliography B
17: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10(vi) Modifications of Expansions in Elementary Functions
… ►The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). … ►Modified Expansions
… ►18: 6.20 Approximations
§6.20(i) Approximations in Terms of Elementary Functions
… ►Cody and Thacher (1968) provides minimax rational approximations for , with accuracies up to 20S.
Cody and Thacher (1969) provides minimax rational approximations for , with accuracies up to 20S.
MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for , , (valid near the origin), and (valid for large ); approximate errors are given for a selection of -values.
19: 28.8 Asymptotic Expansions for Large
Barrett’s Expansions
… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►They are derived by rigorous analysis and accompanied by strict and realistic error bounds. … ►20: Errata
Originally named as a complementary error function, has been renamed as the Faddeeva (or Faddeyeva) function.