The Fermi–Dirac functions. Direct calculation of the functions

The paper presents methods for direct calculation of the Fermi–Dirac functions with a given accuracy. For functions of the integer index, this problem is solved with the help of formula connecting functions of positive and negative arguments. For the function of the half-integer index values of the argument are divided into three areas: negative arguments, where the fast converging series is used; large positive arguments, where asymptotic expansion is used; the intermediate region, where direct numerical integration is used. In the latter case of the constructed formula have exponential ( i.e. very fast) convergence. The properties of such quadrature formulas are investigated. A nontrivial method is found for calculation of integral Fermi–Dirak function. The problem of triple integral calculation comes to calculation of double integral by quadratures with exponential convergence. These methods permit to calculate the Fermi–Dirak functions economically with relative accuracy 10$^{-16}$ for arbitrary values of argument.