The Fermi-Dirac functions. Direct calculation of the functions

Abstract:

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

Keywords:

Fermi-Dirac functions, calculation of functions, exponentialy converging quadratures

Publication language:russian,
pages:29

Research direction:

Mathematical problems and theory of numerical methods