Development of a new concept of polar analytic functions useful in Mellin analysis

In this paper, we develop the concept of polar analyticity introduced in Bardaro C, et al. [A fresh approach to the Paley-Wiener theorem for Mellin transforms and the Mellin-Hardy spaces. Math Nachr. 2017;290:2759–2774] and successfully applied in Mellin analysis and in quadrature of functions defined on the positive real axis (see Bardaro C, et al. [Quadrature formulae for the positive real axis in the setting of Mellin analysis: sharp error estimates in terms of the Mellin distance]. Calcolo. 2018;55(3):26. This appears as a simple way to describe functions which are analytic on a part of the Riemann surface of the logarithm. We study analogues of Cauchy’s integral theorems for polar-analytic functions and obtain two series expansions in terms of polar-derivatives and Mellin polar-derivatives, respectively. We also describe some geometric properties of polaranalytic functions related to conformality. By these studies, we launch the proposal to develop a complete complex function theory, independent of the classical function theory, which is built upon the new notion of polar analyticity.