Abstract
Although the logarithm of a derivative operator ln(Dx) arises naturally in certain contexts, its properties remain relatively unexplored. In this paper, we provide a systematic study of ln(Dx) and demonstrate how it acts on a range of functions, including trigonometric functions and Gaussians. We also derive a unifying formula for the application of ln(Dx) to power series whose coefficients come from an arbitrary analytic function. This leads to several new closed-form evaluations and illustrates the potential of ln(Dx) in operational calculus and the theory of special functions.