auxiliary%20functions
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1: 6.20 Approximations
MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
2: 10.75 Tables
British Association for the Advancement of Science (1937) tabulates , , , 7–8D; , , , 7–10D; , , , , , 8D. Also included are auxiliary functions to facilitate interpolation of the tables of , for small values of .
3: 7.24 Approximations
§7.24(i) Approximations in Terms of Elementary Functions
►Hastings (1955) gives several minimax polynomial and rational approximations for , and the auxiliary functions and .
Cody (1969) provides minimax rational approximations for and . The maximum relative precision is about 20S.
Cody et al. (1970) gives minimax rational approximations to Dawson’s integral (maximum relative precision 20S–22S).
Bulirsch (1967) provides Chebyshev coefficients for the auxiliary functions and for (15D).
4: 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.
5: 12.19 Tables
§12.19 Tables
… ►Miller (1955) includes , , and reduced derivatives for , , 8D or 8S. Modulus and phase functions, and also other auxiliary functions are tabulated.
Fox (1960) includes modulus and phase functions for and , and several auxiliary functions for , , 8S.
Murzewski and Sowa (1972) includes for , , 7S.
6: 6.19 Tables
§6.19(ii) Real Variables
… ►Zhang and Jin (1996, pp. 652, 689) includes , , , 8D; , , , 8S.
Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of , , , 6D; , , , 6D; , , , 6D.
Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of , , , 8S.