§ 5.23. Approximations

Referenced by:: §5.22(i)
Permalink:: http://dlmf.nist.gov/5.23

§5.23(i)Rational Approximations
§5.23(ii)Expansions in Chebyshev Series
§5.23(iii)Approximations in the Complex Plane

§ 5.23(i). Rational Approximations

Keywords:: gamma function, psi function
Permalink:: http://dlmf.nist.gov/5.23.SS1

Cody and Hillstrom (1967) gives minimax rational approximations for $\ln\Gamma\!\left(x\right)$ for the ranges $0.5\le x\le 1.5$ , $1.5\le x\le 4$ , $4\le x\le 12$ ; precision is variable. Hart et al. (1968) gives minimax polynomial and rational approximations to $\Gamma\!\left(x\right)$ and $\ln\Gamma\!\left(x\right)$ in the intervals $0\le x\le 1$ , $8\le x\le 1000$ , $12\le x\le 1000$ ; precision is variable. Cody et al. (1973) gives minimax rational approximations for $\psi\!\left(x\right)$ 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

Keywords:: gamma function, psi function
Permalink:: http://dlmf.nist.gov/5.23.SS2

Luke (1969) gives the coefficients to 20D for the Chebyshev-series expansions of $\Gamma\!\left(1+x\right)$ , $1/\Gamma\!\left(1+x\right)$ , $\Gamma\!\left(x+3\right)$ , $\ln\Gamma\!\left(x+3\right)$ , $\psi\!\left(x+3\right)$ , and the first six derivatives of $\psi\!\left(x+3\right)$ for $0\le x\le 1$ . These coefficients are reproduced in Luke (1975). Clenshaw (1962) also gives 20D Chebyshev-series coefficients for $\Gamma\!\left(1+x\right)$ and its reciprocal for $0\le x\le 1$ . See Luke (1975, pp. 22–23) for additional expansions.