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Evaluation of the Fermi-Dirac integral of half-integer order. (English) Zbl 0758.65012
The author computes $F\sb \mu$ for half-integer $\mu$ and for $z\le 2$, where $F\sb \mu$ is defined by the integral $F\sb \mu(x)=\int\sp \infty\sb 0(1+e\sp{x-2})\sp{-1}x\sp \mu dx$, $\mu>-1$. He also shows that it is possible to evaluate $F\sb \mu(z)$ for half-integer $\mu$ and for sufficiently greater $z$, say for $z\ge 2$, with accuracy only a bit worse than that which is guaranteed by the arithmetic in use. The same methods could be used to evaluate some linear combinations of $F\sb \mu$ important in certain physical applications. Since there are well-known difficulties associated with rational interpolation and approximation, such as nonexistence of an interpolant, rather complicated characterization of the best approximation is required. This phenomenon does not occur in the cases considered here.
65D20Computation of special functions, construction of tables
41A20Approximation by rational functions
30B70Continued fractions (function-theoretic results)
33B20Incomplete beta and gamma functions