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Development of a new concept of polar analytic functions useful in Mellin analysis. (English) Zbl 1428.30003

Summary: In this paper, we develop the concept of polar analyticity introduced in [the authors, Math. Nachr. 290, No. 17–18, 2759–2774 (2017; Zbl 1391.44004)] and successfully applied in Mellin analysis and in quadrature of functions defined on the positive real axis (see [the authors, Calcolo 55, No. 3, Paper No. 26, 33 p. (2018; Zbl 1401.41020)]). 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 polar-analytic 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.

MSC:

30B10 Power series (including lacunary series) in one complex variable
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30C20 Conformal mappings of special domains
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