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Chebyshev-series expansions

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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
  • Clenshaw (1962) gives Chebyshev coefficients for - E 1 ( x ) - ln | x | for - 4 x 4 and e x E 1 ( x ) for x 4 (20D).

  • Luke and Wimp (1963) covers Ei ( x ) for x - 4 (20D), and Si ( x ) and Ci ( x ) for x 4 (20D).

  • 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: 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 .

  • 5: 13.31 Approximations
    §13.31(i) Chebyshev-Series Expansions
    Luke (1969b, pp. 35 and 25) provides Chebyshev-series expansions of M ( a , b , x ) and U ( a , b , x ) that include the intervals 0 x α and α x < , respectively, where α is an arbitrary positive constant. …
    6: 9.19 Approximations
    §9.19(ii) Expansions in Chebyshev Series
  • MacLeod (1994) supplies Chebyshev-series expansions to cover Gi ( x ) for 0 x < and Hi ( x ) for - < x 0 . The Chebyshev coefficients are given to 20D.

  • 7: 25.20 Approximations
  • Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of s ζ ( s + 1 ) and ζ ( s + k ) , k = 2 , 3 , 4 , 5 , 8 , for 0 s 1 (23D).

  • Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover ζ ( s ) for 0 s 1 (15D), ζ ( s + 1 ) for 0 s 1 (20D), and ln ξ ( 1 2 + i x ) 25.4) for - 1 x 1 (20D). For errata see Piessens and Branders (1972).

  • 8: 19.38 Approximations
    Cody (1965b) gives Chebyshev-series expansions3.11(ii)) with maximum precision 25D. …
    9: 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.

  • 10: 5.23 Approximations
    §5.23(ii) Expansions in Chebyshev Series
    Luke (1969b) gives the coefficients to 20D for the Chebyshev-series expansions of Γ ( 1 + x ) , 1 / Γ ( 1 + x ) , Γ ( x + 3 ) , ln Γ ( x + 3 ) , ψ ( x + 3 ) , and the first six derivatives of ψ ( x + 3 ) for 0 x 1 . …