Fourier Transform of some irrational integrands

TL;DR

This paper derives explicit Fourier sine transform formulas for irrational integrands, linking them to Lommel functions and Fresnel integrals, with applications in semi-classical atomic physics.

Contribution

It provides new closed-form expressions for Fourier transforms of irrational functions, connecting them to special functions and physical applications.

Findings

Explicit formulas involving Lommel functions and Fresnel integrals

Applications to semi-classical theory of atomic chemical potential

Enhanced understanding of irrational integrand transforms

Abstract

Fourier sine transforms containing irrational integrands are presented. Explicit closed form expressions are shown to be related to Lommel functions and in special cases to the Fresnel integrals. Such integrals arise in the semi-classical theory of the Chemical Potential of the atomic elements.