  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.
