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§5.23 Approximations


§5.23(i) Rational Approximations

Cody and Hillstrom (1967) gives minimax rational approximations for lnΓ(x) for the ranges 0.5x1.5, 1.5x4, 4x12; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to Γ(x) and lnΓ(x) in the intervals 0x1, 8x1000, 12x1000; precision is variable. Cody et al. (1973) gives minimax rational approximations for ψ(x) for the ranges 0.5x3 and 3x<; precision is variable.

For additional approximations see Hart et al. (1968, Appendix B), Luke (1975, pp. 22–23), and Weniger (2003).

§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), lnΓ(x+3), ψ(x+3), and the first six derivatives of ψ(x+3) for 0x1. These coefficients are reproduced in Luke (1975). Clenshaw (1962) also gives 20D Chebyshev-series coefficients for Γ(1+x) and its reciprocal for 0x1. See Luke (1975, pp. 22–23) for additional expansions.

§5.23(iii) Approximations in the Complex Plane

See Schmelzer and Trefethen (2007) for a survey of rational approximations to various scaled versions of Γ(z).

For rational approximations to ψ(z)+γ see Luke (1975, pp. 13–16).