Fermi-Dirac functions of integral order

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, F$_{n}$($\eta $) = $\int_{0}^{\infty}$ {x$^{n}$/(e$^{x-\eta}$+ 1)} dx, for positive integral values of n. The following relationship is derived: F$_{n}$(+$|\eta|$)+(-)$^{n+1}$ F$_{n}$(-$|\eta|$) = S$_{n}$(+$|\eta|$), where S$_{n}$($\eta $) = $\frac{\eta ^{n+1}}{n+1}\left\{1+\sum_{r=1}2(n+1)\, n\ldots (n-2r+2)\,(1-2^{1-2r})\,\zeta (2r)\,\eta ^{-2r}\right\}$; and the expressions for S$_{n}$($\eta $) are tabulated for n= 1, 2, 3, 4. A series suitable for the evaluation of F$_{n}$(-$|\eta|$) to any required accuracy is indicated; together with the derived relationship this provides a means by which F$_{n}$(+ $|\eta|$) may be computed to any required accuracy. To facilitate the use of the functions the values of (1/n!) F$_{n}$(- $|\eta|$) for n= 1, 2, 3, 4 have been calculated and are tabulated to seven decimal places for $\eta $ = 0$\cdot $0(0$\cdot $1) 4$\cdot $0.