The Analytical Evaluation of the Half-Order Fermi-Dirac Integrals

ABSTRACT

This paper presents a derivation for analytically evaluating the half-order Fermi-Dirac integrals. A complete analytical derivation of the Fermi-Dirac integral of order 1/2 is developed and then generalized to yield each half-order Femi-Dirac function. The most important step in evaluating the Fermi-Dirac integral is to rewrite the integral in terms of two convergent real convolution integrals. Once this done, the Fermi-Dirac integral can put into a form in which a proper contour of integration can be chosen in the complex plane. The application of the theorem of residues reduces the Fermi-Dirac integral into one which becomes analytically tractable. The final solution is written in terms of the complementary and imaginary Error functions.