Journal of Complex Analysis

Volume 2018, Article ID 5941485, 8 pages

https://doi.org/10.1155/2018/5941485

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

^{1}Private Practice, Schenectady, NY 12308, USA^{2}Private Practice, Pittsburgh, PA 15217, USA

Correspondence should be addressed to Jerry P. Selvaggi; moc.liamg@44tsspj

Received 13 September 2017; Revised 6 March 2018; Accepted 20 March 2018; Published 6 May 2018

Academic Editor: N. K. Govil

Copyright © 2018 Jerry P. Selvaggi and Jerry A. Selvaggi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 1. Introduction

This article presents an extension to the method developed by J. A. Selvaggi and J. P. Selvaggi [1] for analytically evaluating* Fermi-Dirac-type* and* Bose-Einstein-type integrals*. The Fermi-Dirac and Bose-Einstein integrals occupy an important role in areas such as solid-state physics and statistical mechanics. There exists numerous approximate analytical methods for evaluating these integrals, but none of the published methods give a general technique for analytically evaluating these integrals valid for the full range of the degeneracy parameter. This article illustrates a general method for attacking these integrals using real convolution.

The right-hand side of (1) and (2) will be defined as* Fermi-Dirac-type integral* and the* Bose-Einstein-type integral*, respectively.Analytical evaluation of these integrals yields the functions and . These functions will be defined as the* Fermi-Dirac function* and the* Bose-Einstein function*, respectively. The parameter, , is called the degeneracy parameter, a term encountered in statistical mechanics. However, as far as the integrals in (1) and (2) are concerned, represents any real number. In general, integrals of this type* do not* allow for closed-form solutions in terms of elementary functions. This article introduces a general method for analytically evaluating the integrals given in (1) and (2) for various functions, . The denominator of the integrands in (1) or (2) is exactly that found in the familiar Fermi-Dirac [2–4] or Bose-Einstein integrals [5, 6]. These integrals are often encountered in statistical and quantum statistical mechanics [7–9]. The authors will mainly consider the Fermi-Dirac functions and Bose-Einstein functions within that domain for which . If , (1) and (2) may be solved by elementary methods.

Numerous techniques have been employed to analytically approximate and numerically evaluate the half-order Fermi-Dirac functions [2, 10–18] and half-order Bose-Einstein functions [5, 17, 18]* (see Appendix)*. A relatively new representation for these integrals, for , is the Polylogarithm function [19–22]. This function has been studied extensively in the literature. In fact, mathematical software such as Mathematica [23] uses a Polylogarithm algorithm to numerically compute the Fermi-Dirac and Bose-Einstein integrals. However, 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, . 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 Fermi-Dirac functions where . A few examples will be considered which will help to illustrate the efficacy of the method. Each solution was numerically checked by employing Mathematica [23] and other numerical algorithms.

#### 2. Theoretical Development

Rewrite the integrals given in (1) and (2) in terms of two convergent real-convolution integrals [24]. The observation that real convolution can be employed is the main focus of this article. In fact, once the integrals are transformed into the proper real-convolution form, the difficulty in analytically evaluating (1) and (2) may be substantially reduced.

In order to see that (1) and (2) can each be transformed into two convergent real-convolution integrals requires that each integral be put into the proper form. To this end, rewrite (1) and (2) as follows:valid .

The denominators in each of the integrals in (3) and (4) can be expanded in a binomial expansion. Each expansion results in a convergent integral within its limits of integration. Employing (3) and expanding the denominator of both integrals in their appropriate binomial expansion yieldsLikewise, (4) can be rewritten as

Depending upon the complexity of the function, , direct integration may be possible. However, the authors will develop the necessary mathematical machinery employing the Laplace Transform inversion integral and contour integration of complex variable theory in order to analytically evaluate (5) and (6). The two integrals within the summation in (5), for example, are real-convolution integrals [24] defined as follows:where is defined as the Laplace Transform of . Both expressions are valid for One can eventually relax the restriction on to include . Substituting (7) into (5) yields the following:Equation (8) represents an exact alternative expression for (1). The only restriction put upon is that it must be a function which allows the integral in (1) to be convergent. Of course, must have a Laplace Transform. The same procedure illustrated above yields the following expression for the Bose-Einstein-type integral.

The first integral, shown in (8) or (9), may be quite simple for practical problems if is an analytically integrable function. The second integral, shown in (8) or (9), is the inverse Laplace Transform [24–26] defined as follows:where

The contour defined by the limits of integration in (10a) is called the* Bromwich Contour* [24–26]. Of course, there is no guarantee that the analytical evaluation of the inversion integral of (10a) will be an easy task. However, for many practical problems found in the literature, this appears not to be a problem.

#### 3. Application of Real Convolution

##### 3.1. Example 1

This simple example is used to verify the method developed in Section 2. Let and evaluate the expression for the Fermi-Dirac function by employing (8). The result is as follows:

Evaluate the inversion integral in by applying the theory of residues of complex variable theory. This is accomplished by choosing the appropriate contour in the complex plane. Figure 1 represents just one possible contour chosen to evaluate the integral in . The contour is closed in the left-hand plane in order to ensure convergence.