Fermi-Dirac functions of integral order

Abstract

Abstract After a brief indication of the types of physical problem in which they arise, an account is given of methods of evaluation of the Fermi-Dirac functions, Fn(ƞ) = ∫0∞ {xn/ez-ƞ + 1)} dx, for positive integral values of n. The following relationship is derived: Fn( + | ƞ |) + (-)n+1Fn(- | ƞ |) = Sn( + | ƞ |), where Sn(ƞ) = (ƞn+1)/(n+1) {1 + ∑ 2(n + 1)n...(n - 2r + 2) (1 - 21-2r) ζ(2r)ƞ-2r}; and the expressions for Sn(ƞ) are tabulated for n = 1,2, 3, 4. A series suitable for the evaluation of Fn( - | ƞ |) to any required accuracy is indicated; together with the derived relationship this provides a means by which Fn( + | ƞ |) may be computed to any required accuracy. To facilitate the use of the functions the values of (1/n!) Fn( - | ƞ |) for n = 1, 2, 3, 4 have been calculated and are tabulated to seven decimal places for ƞ = 0.0(0.1) 4.0.