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1: 4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
2: 6.20 Approximations
§6.20(ii) Expansions in Chebyshev Series
  • 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, pp. 321–322) covers Ein ( x ) and Ein ( x ) for 0 x 8 (the Chebyshev coefficients are given to 20D); E 1 ( x ) for x 5 (20D), and Ei ( x ) for x 8 (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

  • 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.

  • 3: 12.20 Approximations
    §12.20 Approximations
    Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions U ( a , b , x ) and M ( a , b , x ) 13.2(i)) whose regions of validity include intervals with endpoints x = and x = 0 , respectively. As special cases of these results a Chebyshev-series expansion for U ( a , x ) valid when λ x < follows from (12.7.14), and Chebyshev-series expansions for U ( a , x ) and V ( a , x ) valid when 0 x λ follow from (12.4.1), (12.4.2), (12.7.12), and (12.7.13). …
    4: 12.18 Methods of Computation
    These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
    5: 8.27 Approximations
  • Luke (1969b, pp. 25, 40–41) gives Chebyshev-series expansions for Γ ( a , ω z ) (by specifying parameters) with 1 ω < , and γ ( a , ω z ) with 0 ω 1 ; see also Temme (1994b, §3).

  • Luke (1975, p. 103) gives Chebyshev-series expansions for E 1 ( x ) and related functions for x 5 .

  • 6: 13.24 Series
    §13.24 Series
    §13.24(i) Expansions in Series of Whittaker Functions
    For expansions of arbitrary functions in series of M κ , μ ( z ) functions see Schäfke (1961b).
    §13.24(ii) Expansions in Series of Bessel Functions
    For other series expansions see Prudnikov et al. (1990, §6.6). …
    7: 8.25 Methods of Computation
    §8.25(i) Series Expansions
    Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of z , they are cumbersome to use when | z | is large owing to slowness of convergence and cancellation. …
    8: 11.15 Approximations
    §11.15(i) Expansions in Chebyshev Series
  • Luke (1975, pp. 416–421) gives Chebyshev-series expansions for 𝐇 n ( x ) , 𝐋 n ( x ) , 0 | x | 8 , and 𝐇 n ( x ) Y n ( x ) , x 8 , for n = 0 , 1 ; 0 x t m 𝐇 0 ( t ) d t , 0 x t m 𝐋 0 ( t ) d t , 0 | x | 8 , m = 0 , 1 and 0 x ( 𝐇 0 ( t ) Y 0 ( t ) ) d t , x t 1 ( 𝐇 0 ( t ) Y 0 ( t ) ) d t , x 8 ; the coefficients are to 20D.

  • MacLeod (1993) gives Chebyshev-series expansions for 𝐋 0 ( x ) , 𝐋 1 ( x ) , 0 x 16 , and I 0 ( x ) 𝐋 0 ( x ) , I 1 ( x ) 𝐋 1 ( x ) , x 16 ; the coefficients are to 20D.

  • 9: 12.4 Power-Series Expansions
    §12.4 Power-Series Expansions
    10: 24.8 Series Expansions
    §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 , .