Some integrals related to the Fermi function

TL;DR

This paper derives analytical expressions for the Hilbert and Fourier transforms of the Fermi function and related integrals, which are crucial for understanding electronic transport properties.

Contribution

It provides new analytical formulas for the Hilbert and Fourier transforms of the Fermi function, including an integral involving their product, advancing theoretical tools in condensed matter physics.

Findings

Hilbert transform expressed via digamma function

Fourier transform in terms of elementary functions

Analytical evaluation of a key integral involving Fermi functions

Abstract

Some elaborations regarding the Hilbert and Fourier transforms of Fermi function are presented. The main result shows that the Hilbert transform of the difference of two Fermi functions has an analytical expression in terms of the $\Psi$ (digamma) function, while its Fourier transform is expressed by mean of elementary functions. Moreover an integral involving the product of the difference of two Fermi functions with its Hilbert transform is evaluated analytically. These findings are of fundamental importance in discussing the transport properties of electronic systems.