Fourier Transform Representation of the Extended Fermi-Dirac and Bose-Einstein Functions with Applications to the Family of the Zeta and Related Functions

TL;DR

This paper develops Fourier transform methods to analyze extended Fermi-Dirac and Bose-Einstein functions, deriving new integral formulas involving zeta functions and advancing the mathematical understanding of these special functions.

Contribution

It introduces Fourier transform representations for extended Fermi-Dirac and Bose-Einstein functions, enabling the evaluation of previously unknown integrals involving zeta functions.

Findings

Derived new integral formulas involving Riemann and Hurwitz zeta functions.

Evaluated integrals of products of extended Fermi-Dirac and Bose-Einstein functions.

Provided mathematical tools linking special functions with Fourier analysis.

Abstract

On the one hand the Fermi-Dirac and Bose-Einstein functions have been extended in such a way that they are closely related to the Riemann and other zeta functions. On the other hand the Fourier transform representation of the gamma and generalized gamma functions proved useful in deriving various integral formulae for these functions. In this paper we use the Fourier transform representation of the extended functions to evaluate integrals of products of these functions. In particular we evaluate some integrals containing the Riemann and Hurwitz zeta functions, which had not been evaluated before.