The Application of Real Convolution for Analytically Evaluating Fermi-Dirac-Type and Bose-Einstein-Type Integrals

Jerry P. Selvaggi,Jerry A. Selvaggi

doi:10.1155/2018/5941485

Abstract

The Fermi-Dirac-type or Bose-Einstein-type integrals can be transformed into two convergent real-convolution integrals. The transformation simplifies the integration process and may ultimately produce a complete analytical solution without recourse to any mathematical approximations. The real-convolution integrals can either be directly integrated or be transformed into the Laplace Transform inversion integral in which case the full power of contour integration becomes available. Which method is employed is dependent upon the complexity of the real-convolution integral. A number of examples are introduced which will illustrate the efficacy of the analytical approach.

Highlights

This article presents an extension to the method developed by J
The FermiDirac and Bose-Einstein integrals occupy an important role in areas such as solid-state physics and statistical mechanics
Analytical evaluation of these integrals yields the functions F(ξ) and B(ξ). These functions will be defined as the FermiDirac function and the Bose-Einstein function, respectively

Summary

Introduction

This article presents an extension to the method developed by J. P. Selvaggi [1] for analytically evaluating Fermi-Dirac-type and Bose-Einstein-type integrals. Analytical evaluation of these integrals yields the functions F(ξ) and B(ξ) These functions will be defined as the FermiDirac function and the Bose-Einstein function, respectively. This article introduces a general method for analytically evaluating the integrals given in (1) and (2) for various functions, f(x). The authors will illustrate that the application of real convolution allows for the complete analytical evaluation of the integrals in (1) and (2) for a wide range of functions, f(x). The authors have already employed this technique [1] to analytically evaluate the integral in (1) for the well-known and important case of the half-order FermiDirac functions where f(x) = xm−1/2 ∀m ∈ Z≥. Each solution was numerically checked by employing Mathematica [23] and other numerical algorithms

Theoretical Development

Application of Real Convolution

Remarks and Conclusion