Precise and fast computation of Fermi–Dirac integral of integer and half integer order by piecewise minimax rational approximation Toshio Fukushima National Astronomical Observatory of Japan, Graduate University of General Sciences, 2-21-1, Ohsawa, Mitaka, Tokyo 181-8588, Japan article info Keywords: Chebyshev series expansion Fermi–Dirac integral Maclaurin series expansion Minimax rational approximation Sommerfeld expansion abstract Piecewise quadruple precision approximations of the Fermi–Dirac integral of integer order, 0(1)10, and half integer order, 9=2ð1Þ21=2, are developed by combining (i) the optimally truncated Sommerfeld expansion, (ii) the piecewise truncated Chebyshev series expansion, and/or (iii) the reflection formula. They are used in constructing the double precision piece- wise minimax rational approximations of the integral of the same orders. The relative errors of the new minimax approximations, which are all due to rounding off, are 3–13 machine epsilons at most and less than 4 machine epsilons typically while their CPU times are only 0.44–1.1 times that of the exponential function when 5 6 g 6 45. As a result, the new approximations run 16–31 times faster than the piecewise Chebyshev polynomial approximations (Macleod, 1998) for the physically important orders, 1=2ð1Þ5=2, and 330–720 and 50–93 times faster than the combined series expansions (Goano, 1995) for the half integer orders, 1=2ð1Þ21=2, and the integer orders, 1(1)9, respectively. A file of the Fortran codes of the obtained approximations and their test program and sample out- put is named xfdh.txt and located at: https://www.researchgate.net/profile/Toshio_ Fukushima/. Ó 2015 Elsevier Inc. All rights reserved. 1. Introduction 1.1. Fermi–Dirac integral of integer and half integer order The Fermi–Dirac integral is an important special function in quantum statistics, especially in the solid state physics [1]. It is written in the form of an integral transform of a power function [2, Eq. (1.15)] as F k ðgÞ Z 1 0 x k pðx  gÞdx; ðk > 1; 1 < g < 1Þ; ð1Þ where the kernel of the transform pðzÞ 1 expðzÞþ 1 ; ð2Þ http://dx.doi.org/10.1016/j.amc.2015.03.009 0096-3003/Ó 2015 Elsevier Inc. All rights reserved. E-mail address: Toshio.Fukushima@nao.ac.jp Applied Mathematics and Computation 259 (2015) 708–729 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc