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1: 24.8 Series Expansions
§24.8(i) Fourier Series
If n = 1 , 2 , and 0 x 1 , then …
§24.8(ii) Other Series
24.8.9 E 2 n = ( - 1 ) n k = 1 k 2 n cosh ( 1 2 π k ) - 4 k = 0 ( - 1 ) k ( 2 k + 1 ) 2 n e 2 π ( 2 k + 1 ) - 1 , n = 1 , 2 , .
2: 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
22.12.13 2 K cs ( 2 K t , k ) = lim N n = - N N ( - 1 ) n π tan ( π ( t - n τ ) ) = lim N n = - N N ( - 1 ) n ( lim M m = - M M 1 t - m - n τ ) .
3: 25.2 Definition and Expansions
§25.2(ii) Other Infinite Series
25.2.5 γ n = lim m ( k = 1 m ( ln k ) n k - ( ln m ) n + 1 n + 1 ) .
4: Bibliography S
  • H. Shanker (1940a) On integral representation of Weber’s parabolic cylinder function and its expansion into an infinite series. J. Indian Math. Soc. (N. S.) 4, pp. 34–38.
  • 5: 31.11 Expansions in Series of Hypergeometric Functions
    §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. …
    6: 25.12 Polylogarithms
    The cosine series in (25.12.7) has the elementary sum … For real or complex s and z the polylogarithm Li s ( z ) is defined by … For each fixed complex s the series defines an analytic function of z for | z | < 1 . The series also converges when | z | = 1 , provided that s > 1 . … The notation ϕ ( z , s ) was used for Li s ( z ) in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function. …
    7: 16.11 Asymptotic Expansions
    8: 18.38 Mathematical Applications
    In consequence, expansions of functions that are infinitely differentiable on [ - 1 , 1 ] in series of Chebyshev polynomials usually converge extremely rapidly. …
    9: 2.1 Definitions and Elementary Properties
    In those cases it is usually necessary to interpret each infinite series separately in the manner described above; that is, it is not always possible to reinterpret the asymptotic approximation as a single asymptotic expansion. …
    10: 28.11 Expansions in Series of Mathieu Functions
    §28.11 Expansions in Series of Mathieu Functions
    Let f ( z ) be a 2 π -periodic function that is analytic in an open doubly-infinite strip S that contains the real axis, and q be a normal value (§28.7). …The series (28.11.1) converges absolutely and uniformly on any compact subset of the strip S . See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of q see Meixner et al. (1980, p. 33). …
    28.11.7 sin ( 2 m + 2 ) z = n = 0 B 2 m + 2 2 n + 2 ( q ) se 2 n + 2 ( z , q ) .