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trigonometric series expansions

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1: 4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
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: 22.11 Fourier and Hyperbolic Series
§22.11 Fourier and Hyperbolic Series
4: 3.11 Approximation Techniques
In fact, (3.11.11) is the Fourier-series expansion of f ( cos θ ) ; compare (3.11.6) and §1.8(i). …
5: 6.20 Approximations
  • Luke and Wimp (1963) covers Ei ( x ) for x - 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • Luke (1969b, pp. 41–42) gives Chebyshev expansions of Ein ( a x ) , Si ( a x ) , and Cin ( a x ) for - 1 x 1 , a . The coefficients are given in terms of series of Bessel functions.

  • Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric U -function (§13.2(i)) from which Chebyshev expansions near infinity for E 1 ( z ) , f ( z ) , and g ( z ) follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the U functions. If | ph z | < π the scheme can be used in backward direction.

  • 6: 6.18 Methods of Computation
    Power series, asymptotic expansions, and quadrature can also be used to compute the functions f ( z ) and g ( z ) . …
    7: 8.21 Generalized Sine and Cosine Integrals
    §8.21(vi) Series Expansions
    Power-Series Expansions
    Spherical-Bessel-Function Expansions
    For (8.21.16), (8.21.17), and further expansions in series of Bessel functions see Luke (1969b, pp. 56–57). … For the corresponding expansions for si ( a , z ) and ci ( a , z ) apply (8.21.20) and (8.21.21). …
    8: 19.5 Maclaurin and Related Expansions
    §19.5 Maclaurin and Related Expansions
    Series expansions of F ( ϕ , k ) and E ( ϕ , k ) are surveyed and improved in Van de Vel (1969), and the case of F ( ϕ , k ) is summarized in Gautschi (1975, §1.3.2). For series expansions of Π ( ϕ , α 2 , k ) when | α 2 | < 1 see Erdélyi et al. (1953b, §13.6(9)). …
    9: 24.8 Series Expansions
    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 , .
    10: 28.23 Expansions in Series of Bessel Functions
    §28.23 Expansions in Series of Bessel Functions
    valid for all z when j = 1 , and for z > 0 and | cosh z | > 1 when j = 2 , 3 , 4 . … When j = 1 , each of the series (28.23.6)–(28.23.13) converges for all z . When j = 2 , 3 , 4 the series in the even-numbered equations converge for z > 0 and | cosh z | > 1 , and the series in the odd-numbered equations converge for z > 0 and | sinh z | > 1 . …