On the Logarithm of a Derivative Operator and Its Applications

Abstract Although the logarithm of a derivative operator ln( D x ) arises naturally in certain contexts, its properties remain relatively unexplored. In this paper, we provide a systematic study of ln( D x ) 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( D x ) 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( D x ) in operational calculus and the theory of special functions.