Calculation of the Fermi–Dirac functions with exponentially convergent quadratures

The special quadrature formulas of high accuracy were built for a direct calculation of the Fermi–Dirac functions of half-integer indexes. It is shown that the dependence of the error from the number of nodes is not a power law, but exponential. We investigated the properties of such formulas. It is shown that the index of this exponent is proportional to the distance between the integral segment and the nearest pole of expanded expression. This provides a very fast convergence of quadratures. The simple approximations of the Fermi–Dirac functions of integer and halfinteger indexes were constructed; their accuracy was about 1%. They are convenient for the physical estimations. During the research, we found an asymptotic representation for Bernoulli numbers.