Operator Representation of Fermi-Dirac and Bose-Einstein Integral Functions with Applications

M Aslam Chaudhry,Asghar Qadir

doi:10.1155/2007/80515

Abstract

Fermi-Dirac and Bose-Einstein functions arise as quantum statistical distributions. The Riemann zeta function and its extension, the polylogarithm function, arise in the theory of numbers. Though it might not have been expected, these two sets of functions belong to a wider class of functions whose members have operator representations. In particular, we show that the Fermi-Dirac and Bose-Einstein integral functions are expressible as operator representations in terms of themselves. Simpler derivations of previously known results of these functions are obtained by their operator representations.

Highlights

The study of analytic functions is very useful for the application of mathematics to various physical and engineering problems and for the development of a further understanding of mathematics itself
The present formulation of the Weyl transform representation of the FermiDirac integral functions leads to the representation (see (2.6))
We hope that the present formulation of the operator representation of the Fermi integrals may lead to the desired formula

Summary

Introduction

The study of analytic functions is very useful for the application of mathematics to various physical and engineering problems and for the development of a further understanding of mathematics itself. The Riemann zeta function [1, page 1]. Of special interest for our purposes is the polylogarithm function φ(x, s). It has been studied extensively by several authors including Lambert, Legendre, Abel, Kummer, Appell, Lerch, Lindelof, Wirtinger, Jonquiere, Truesdell, and others. We present a series representation of a class of functions and deduce the well-known series representation and operator forms of the Fermi-Dirac (and Bose-Einstein) integral and other related functions. The present formulation helps us to find an alternate proof of the Euler formula for the closed-form representation of the zeta function at even integral values. The function (1.2) is related to the Fermi-Dirac integral function [4, page 30]. Which is useful in translating the properties of these functions

The Mellin and Weyl transform representations

A class of good functions and applications

Applications to Fermi-Dirac and Bose-Einstein integral functions

Alternate derivation of Euler’s formula

Concluding remarks