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

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.