# §5.23 Approximations

## §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 $\mathop{\psi\/}\nolimits\!\left(z\right)+\gamma$ see Luke (1975, pp. 13–16).