Fermi-Dirac integrals in terms of Zeta Functions

TL;DR

This paper expresses Fermi-Dirac integrals using Riemann and Hurwitz Zeta functions by introducing an auxiliary function, providing a non-iterative, generalized approach that complements Sommerfeld's lemma.

Contribution

It introduces a novel method to represent Fermi-Dirac integrals in terms of Zeta functions, avoiding iterative techniques and extending the applicability within a generalized interval.

Findings

Fermi-Dirac integrals are expressed via Zeta functions.

The method avoids iterative integral calculations.

The approach generalizes and complements Sommerfeld's lemma.

Abstract

This paper shows the Fermi-Dirac Integrals expressed in terms of Riemann and Hurwitz Zeta functions. This is done by defining an auxiliar function that permits rewrite the Fermi-Dirac integral in terms of simpler and known integrals resulting in the Zeta functions mentioned. The approach used here evades the use of iterative methods for the integrals and presents a generalization in a refereed interval, one that complements Sommerfeld lemma.