New Fourier Transform Containing a Pair of Complex Euler Gamma Functions With a Monomial: Mathematical and Physical Applications

TL;DR

This paper introduces a new Fourier transform involving complex gamma functions and monomials, providing a general formula with applications in mathematical analysis and quantum physics, particularly for systems with position-dependent mass.

Contribution

It proposes an elementary method to derive a general Fourier transform formula involving complex gamma functions and applies it to quantum systems with position-dependent effective mass.

Findings

Derived a general Fourier transform formula involving complex gamma functions and hypergeometric functions.

Collected mathematical results based on the new transform.

Applied the transform to compute expectation values in quantum systems with position-dependent mass.

Abstract

One of the goals of the present paper is to propose an elementary method to find a general formula for the Fourier transform containing a pair of complex gamma functions with a monomial sm in terms of Gauss's hypergeometric functions 2F1. We further collect some mathematical results that follow by means of this transform. Physical applications requiring the expectation values of position and momentum operators for a quantum system endowed with position-dependent effective mass are presented.