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1: 28.16 Asymptotic Expansions for Large
2: 20 Theta Functions
Chapter 20 Theta Functions
…3: 6.20 Approximations
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.
4: 10.75 Tables
Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Bickley et al. (1952) tabulates or , or , , (.01 or .1) 10(.1) 20, 8S; , , , or , 10S.
Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of and , for , 9S.
Olver (1960) tabulates , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as .
Zhang and Jin (1996, p. 322) tabulates , , , , , , , , , 7S.
5: Bibliography N
6: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►§28.8(ii) Sips’ Expansions
… ►Barrett’s Expansions
… ►Dunster’s Approximations
… ►7: Gergő Nemes
8: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
… ►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). … ►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
… ►9: 7.24 Approximations
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).
Schonfelder (1978) gives coefficients of Chebyshev expansions for on , for on , and for on (30D).
Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for , , , , and ; approximate errors are given for a selection of -values.