Analytical computation of generalized Fermi–Dirac integrals by truncated Sommerfeld expansions

Abstract

For the generalized Fermi–Dirac integrals, Fk(η,β), of orders k=−1/2, 1/2, 3/2, and 5/2, we explicitly obtained the first 11 terms of their Sommerfeld expansions. The main terms of the last three orders are rewritten so as to avoid the cancelation problem. If η is not so small, say not less than 13.5, 12.0, 10.9, and 9.9 when k=−1/2, 1/2, 3/2, and 5/2, respectively, the first 8 terms of the expansion assure the single precision accuracy for arbitrary value of β. Similarly, the 15-digits accuracy is achieved by the 11 terms expansion if η is greater than 36.8, 31.6, 30.7, and 26.6 when k=−1/2, 1/2, 3/2, and 5/2, respectively. Since the truncated expansions are analytically given in a closed form, their computational time is sufficiently small, say at most 4.9 and 6.7 times that of the integrand evaluation for the 8- and 11-terms expansions, respectively. When η is larger than a certain threshold value as indicated, these appropriately-truncated Sommerfeld expansions provide a factor of 10–80 acceleration of the computation of the generalized Fermi–Dirac integrals when compared with the direct numerical quadrature.