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1: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
►With and …The double sums in (22.12.2)–(22.12.4) are convergent but not absolutely convergent, hence the order of the summations is important. … ►2: 5.19 Mathematical Applications
3: 31.11 Expansions in Series of Hypergeometric Functions
§31.11 Expansions in Series of Hypergeometric Functions
… ►Series of Type II (§31.11(iv)) are expansions in orthogonal polynomials, which are useful in calculations of normalization integrals for Heun functions; see Erdélyi (1944) and §31.9(i). … ►§31.11(v) Doubly-Infinite Series
►Schmidt (1979) gives expansions of path-multiplicative solutions (§31.6) in terms of doubly-infinite series of hypergeometric functions.4: 28.19 Expansions in Series of Functions
§28.19 Expansions in Series of Functions
►Let be a normal value (§28.12(i)) with respect to , and be a function that is analytic on a doubly-infinite open strip that contains the real axis. … ►where the coefficients are as in §28.14.5: 28.11 Expansions in Series of Mathieu Functions
§28.11 Expansions in Series of Mathieu Functions
►Let be a -periodic function that is analytic in an open doubly-infinite strip that contains the real axis, and be a normal value (§28.7). …See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of see Meixner et al. (1980, p. 33). … ►6: 1.9 Calculus of a Complex Variable
7: 20 Theta Functions
Chapter 20 Theta Functions
…8: 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
… ►§6.20(iii) Padé-Type and Rational 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).
§7.24(ii) Expansions in Chebyshev Series
… ►Schonfelder (1978) gives coefficients of Chebyshev expansions for on , for on , and for on (30D).
§7.24(iii) Padé-Type Expansions
…10: 10.75 Tables
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 (1984c) tabulates all zeros of and in the sector 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 .