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1: 4.47 Approximations
§4.47(i) Chebyshev-Series Expansions
2: 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). …
3: 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.

  • 4: 25.20 Approximations
  • Cody et al. (1971) gives rational approximations for ζ ( s ) in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are 0.5 s 5 , 5 s 11 , 11 s 25 , 25 s 55 . Precision is varied, with a maximum of 20S.

  • 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).

  • Antia (1993) gives minimax rational approximations for Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for the intervals - < x 2 and 2 x < , with s = - 1 2 , 1 2 , 3 2 , 5 2 . For each s there are three sets of approximations, with relative maximum errors 10 - 4 , 10 - 8 , 10 - 12 .

  • 5: 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 . …Clenshaw (1962) also gives 20D Chebyshev-series coefficients for Γ ( 1 + x ) and its reciprocal for 0 x 1 . …
    6: 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 .

  • 7: 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.

  • 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: 18.40 Methods of Computation
    However, for applications in which the OP’s appear only as terms in series expansions (compare §18.18(i)) the need to compute them can be avoided altogether by use instead of Clenshaw’s algorithm (§3.11(ii)) and its straightforward generalization to OP’s other than Chebyshev. …
    10: 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. …