Refinements of precision approximations of Fermi-Dirak functions of integer indices

Abstract

Fermi-Dirac functions of integer indices are broadly used in problems of electronic transport in dense substances. Polynomial approximations are constructed for their fast computation. Such coefficients are found for functions of index 1, 2, and 3, which provide an error ratio of about 2 × 10^–16 with nine free parameters. In this work, we use the boost::multiprecision library of C++, which allows us to compute with any arbitrary number of digits. The precision of previously obtained relations is improved to ~5 × 10^–18 and the same relation is constructed for the index k = 4. Also, it is shown that simple global relation consisting of a few parameters reasonably describe the order of the value of the functions for all values of the independent variable and can be used for estimations.