The Open Mathematics Journal

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 FermiDirac integral into one which becomes analytically tractable. The final solution is written in terms of the complementary and imaginary Error functions.