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

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