Computing the Fermi−Dirac Functions by Exponentially Convergent Quadratures

Highly accurate specialized quadrature formulas are constructed for directly computing the Fermi−Dirac functions of the half-integer index. It is shown that the dependence of the error on the number of nodes is not power-law but exponential. The properties of such formulas are investigated. It is demonstrated that the factor of the convergence exponent is determined by the distance to the nearest pole of the integrand. This provides a very fast convergence of the quadratures. Simple approximations of the Fermi−Dirac functions of the integer and half-integer indices with an accuracy better than 1% are constructed; these approximations are convenient for physical estimates. In passing, asymptotic representations for Bernoulli numbers are found.