New Fourier transform containing a pair of complex Euler Γ-functions with a monomial: Mathematical and physical applications

One of the goals of this paper is to propose an elementary method to find a general formula for a new Fourier transform containing a pair of complex Euler Γ-functions with a monomial sm in terms of Gauss’s hypergeometric functions 2F1. Some mathematical results have been obtained and discussed using this transform. Furthermore, physical applications in the phase space quantum mechanics requiring the expectation values of position and momentum operators for a quantum system endowed with position-dependent effective mass are presented.