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Fermi-Dirac Functions of Integral Order

P. Rhodes
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 204, No. 1078 (Dec. 22, 1950), pp. 396-405
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/98693
Page Count: 10
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Fermi-Dirac Functions of Integral Order
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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(η ) = ∫0 ∞ {xn/(ex-η+ 1)} dx, for positive integral values of n. The following relationship is derived: Fn(+|η|)+(-)n+1 Fn(-|η|) = Sn(+|η|), where Sn(η ) = $\frac{\eta ^{n+1}}{n+1}\left\{1+\sum_{r=1}2(n+1)\, n\ldots (n-2r+2)\,(1-2^{1-2r})\,\zeta (2r)\,\eta ^{-2r}\right\}$; and the expressions for Sn(η ) are tabulated for n= 1, 2, 3, 4. A series suitable for the evaluation of Fn(-|η|) to any required accuracy is indicated; together with the derived relationship this provides a means by which Fn(+ |η|) may be computed to any required accuracy. To facilitate the use of the functions the values of (1/n!) Fn(- |η|) for n= 1, 2, 3, 4 have been calculated and are tabulated to seven decimal places for η = 0· 0(0· 1) 4· 0.

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