We will show that every nonsingular complex variety is a manifold in the analytic topology, but we wish to show more – that it has the natural structure of a complex manifold. In order to do this, we will need to introduce additional structure on the analytic topology. To allow for singular points, we give a definition more general than that of complex manifolds, mimicking our definitions from ...

In the last few sections, we completely described entire functions through the use of everywhere convergent power series. Our goal for the following sections is to instead consider functions which are analytic almost everywhere and see which results generalize. We shall see that we do get some generalizations (like representations of functions via convergent power series), but unlike the entire...

In his retiring presidential address before the American Mathematical Society (December, 1931), E. R. Hedrick gave a resumé of the theory of non-analytic functions of a complex variable.f We propose in the present note to establish, with regard to these functions, some elementary properties that do not appear in the Hedrick report. (In the course of the work, however, some properties considered...