Analytical expansions for Fermi-Dirac functions

We obtain a fast convergent series expansion for the Fermi-Dirac function Fer (a) for - lO<;;a<;; - 1. We give values of Feria) for £7 = n +! (n = 0,1,·.·,6) with a in the same range.


I. INTRODUCTION
The Fermi-Dirac functions F(7 (a), where £7 is a positive real parameter, is defined for all real numbers a by When £7 is an integer, this integral may be easily evaluated by a power series; a complete discussion of this case is due to Rhodes. I For arbitrary a, there are several expansions depending on the range of values of a. 2--4 The calculation of F,,(a) for a < 0 is needed in many questions of quantum statistical mechanics; for example, to solve the equations of state corresponding to extreme conditions (high pressure and nonzero temperature). Analytical expansions are available in all ranges, except when -lO<;;a<;; -1. Previous evaluations of Fa(a) for this range were made by numerical integration 4 . 5 or by polynomial approximation. 6 In this paper we obtain a fast convergent series expansion of Fa(a) for -lO<;;a<;; -1. This series converges uniformly for leYl < 1, that is, for y < O.
e" + 1 dy. Now I will be calculated as I = 11 + 12 + 13 by dividing the integration interval by the pointsp andp, where 0 <p < la I. Another restriction on the values of p and convenient numerical suggestions will appear later. ' ''Fellow ofCONICET (Consejo Nacional de Investigaciones Cientificas y Tecnicas, Argentina). e"Y = e"Yo L nk(y -YOlk k;;.O k! By replacing successively in (1), taking into account the uniform convergence of the series to exchange the order of integrals and summations, it follows that n k II = L ( -Ij"e ny " L ., xf -I:: ly + lal)"lly -YOlk dy.
The integrals involved in this expression may be evaluated using the formula
-p eY+l Since 0 <p < ja j, the series C~I + lY-1 = ~J£T~ 1)(\:\)" converges uniformly. The same statement holds? for the series l OO (y + jaj)"-I 13 = dy. is uniformly convergent for y > 0, by exchanging the integral and the summation, with the substitution z = n(y + jalJ we obtain when p < 1T, B k being the nonzero Bernoulli numbers. Therefore, arguments used in Sec. lA) apply here, yielding Thus,1 3 can be expressed in terms of incomplete Gamma functions as and

C. Evaluation of 13
Recall that ( _ l)n + le nlal 13 = I a r(£T,n(p + lal))· n>1 n From our numerical investigations we conclude that, in order to achieve a fast convergence, the values ofp andyo may be chosen as follows: As an application, values of FO"(a) for -10~a~ -1 and (J' = n + 1 (n = 0,1, ... ,6) were computed with a maximum relative error of 10-5 • In particular, we have checked the accuracy of all previously tabulated values. 4 • 6 In the course of the computation we have made use of Abramowitz's tables 7 for Bernoulli numbers. The corresponding computing program is to be published elsewhere. s TABLE I. Values of F,,(a) for -lO<:;;a< -1 and a = n + \ with n = 0,1,.",6. ALPHA -1.0