§25.20 Approximations

Keywords:: Fermi–Dirac integrals, Riemann’s $\xi$ -function, Riemann zeta function, dilogarithms
Permalink:: http://dlmf.nist.gov/25.20

Cody et al. (1971) gives rational approximations for $\mathop{\zeta\/}\nolimits\!\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$ , $5\leq s\leq 11$ , $11\leq s\leq 25$ , $25\leq s\leq 55$ . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\mathop{\zeta\/}\nolimits\!\left(s+1\right)$ and $\mathop{\zeta\/}\nolimits\!\left(s+k\right)$ , $k=2,3,4,5,8$ , for $0\leq s\leq 1$ (23D).
Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\mathop{\zeta\/}\nolimits\!\left(s\right)$ for $0\leq s\leq 1$ (15D), $\mathop{\zeta\/}\nolimits\!\left(s+1\right)$ for $0\leq s\leq 1$ (20D), and $\mathop{\ln\/}\nolimits\mathop{\xi\/}\nolimits\!\left(\tfrac{1}{2}+ix\right)$ (§25.4) for $-1\leq x\leq 1$ (20D). For errata see Piessens and Branders (1972).
Morris (1979) gives rational approximations for $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(x\right)$ (§25.12(i)) for $0.5\leq x\leq 1$ . Precision is varied with a maximum of 24S.
Antia (1993) gives minimax rational approximations for $\mathop{\Gamma\/}\nolimits\!\left(s+1\right)F_{s}(x)$ , where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty<x\leq 2$ and $2\leq x<\infty$ , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$ . For each $s$ there are three sets of approximations, with relative maximum errors $10^{{-4}},10^{{-8}},10^{{-12}}$ .