Development of a complex function theory upon a new concept of polar-analytic functions; Extended version

TL;DR

This paper proposes a new complex function theory based on polar analyticity, extending previous work and aiming to describe functions on Riemann surfaces, with applications in Mellin analysis and quadrature formulas.

Contribution

It introduces a comprehensive complex function theory founded on polar analyticity, independent of classical theories, with new results and an updated reference list.

Findings

Successful application in Mellin analysis

Development of quadrature formulas for positive real functions

Extension of polar analyticity to a full complex theory

Abstract

The present article is an extended version of [6] containing new results and an updated list of references. We review the notion of polar analyticity introduced in a previous paper and succesfully applied in Mellin analysis and quadrature formulae for functions defined on the positive real axis. This appears as a simple way to describe functions which are analytic on a part of the Riemann surface of the logarithm. In this paper we launch a proposal to develop a complete complex function theory, independent of classical function theory, which is built upon the new concept of polar analyticity.