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5 Gamma FunctionComputation

§5.23 Approximations

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§5.23(i) Rational Approximations

Cody and Hillstrom (1967) gives minimax rational approximations for \mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(x\right) for the ranges 0.5\leq x\leq 1.5, 1.5\leq x\leq 4, 4\leq x\leq 12; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to \mathop{\Gamma\/}\nolimits\!\left(x\right) and \mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(x\right) in the intervals 0\leq x\leq 1, 8\leq x\leq 1000, 12\leq x\leq 1000; precision is variable. Cody et al. (1973) gives minimax rational approximations for \mathop{\psi\/}\nolimits\!\left(x\right) for the ranges 0.5\leq x\leq 3 and 3\leq x<\infty; 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 \mathop{\Gamma\/}\nolimits\!\left(1+x\right), 1/\mathop{\Gamma\/}\nolimits\!\left(1+x\right), \mathop{\Gamma\/}\nolimits\!\left(x+3\right), \mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(x+3\right), \mathop{\psi\/}\nolimits\!\left(x+3\right), and the first six derivatives of \mathop{\psi\/}\nolimits\!\left(x+3\right) for 0\leq x\leq 1. These coefficients are reproduced in Luke (1975). Clenshaw (1962) also gives 20D Chebyshev-series coefficients for \mathop{\Gamma\/}\nolimits\!\left(1+x\right) and its reciprocal for 0\leq x\leq 1. 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 \mathop{\Gamma\/}\nolimits\!\left(z\right).

For rational approximations to \mathop{\psi\/}\nolimits\!\left(z\right)+\EulerConstant see Luke (1975, pp. 13–16).