Integration of polar-analytic functions and applications to Boas' differentiation formula and Bernstein's inequality in Mellin setting (Q2227252)

This article is a continuation of a series of articles by the same authors in which they introduced the notion of polar analytic functions and derived properties of functions that meet that definition. A function is said to have a polar derivative at \((r_0, \theta_0)\) in the upper half-plane if \[(D_{\operatorname{pol}} f) (r_0, \theta_0)=\lim_{(r,\theta)\rightarrow (r_0,\theta_0)}\frac{ f(r,\theta)-f(r_0, \theta_0) }{ re^{i\theta}-r_0 e^{i\theta_0} } \] exits. A corresponding notion of Mellin polar derivative is defined as \[ \Theta_c f(r, \theta)=re^{i\theta}(D_{\operatorname{pol}} f) (r, \theta)+cf(r, \theta). \] In a previous article an extension of Cauchy's integral formula and Taylor-type series were established among other results. In this article under review the aim is to establish another analogue of Cauchy's integral formula which is free of artifacts and then derive a version of the residue theorem for polar-analytic functions. An analogue of Boas' differentiation formula for polar Mellin derivatives and a Bernstein-type inequality are obtained