Dougall%20expansion
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1: 16.4 Argument Unity
Rogers–Dougall Very Well-Poised Sum
… βΊDougall’s Very Well-Poised Sum
… βΊThe function is analytic in the parameters when its series expansion converges and the bottom parameters are not negative integers or zero. … βΊThis is Dougall’s bilateral sum; see Andrews et al. (1999, §2.8).2: Bibliography D
3: 14.18 Sums
§14.18(i) Expansion Theorem
βΊFor expansions of arbitrary functions in series of Legendre polynomials see §18.18(i), and for expansions of arbitrary functions in series of associated Legendre functions see Schäfke (1961b). … βΊDougall’s Expansion
…4: 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.
§6.20(ii) Expansions in Chebyshev Series
… βΊ§6.20(iii) Padé-Type and Rational Expansions
…5: 28.16 Asymptotic Expansions for Large
6: 20 Theta Functions
Chapter 20 Theta Functions
…7: 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).
§7.24(ii) Expansions in Chebyshev Series
… βΊSchonfelder (1978) gives coefficients of Chebyshev expansions for on , for on , and for on (30D).
§7.24(iii) Padé-Type Expansions
…8: 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).
9: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
… βΊFor re-expansions of the remainder terms in (11.6.1) and (11.6.2), see Dingle (1973, p. 445). … βΊMore fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). … βΊHere …10: 10.75 Tables
Achenbach (1986) tabulates , , , , , 20D or 18–20S.
Olver (1960) tabulates , , , , , , , , , , 8D. Also included are tables of the coefficients in the uniform asymptotic expansions of these zeros and associated values as ; see §10.21(viii), and more fully Olver (1954).
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 .