The main idea of ​​this book is to provide a good part of the standard material on complex variable functions, as well as some new materials, from the point of view of functional analysis. The main objective of the study is algebraic H (G) for all the total functions of the open group G, with the H (G) topology of the uniform convergence of small subsets of G. From this point of vie ~, the main theory of theory is theory 9.5, (H) with the space of germs and holomorphic functions on the G.

supplement of this result, for example, Runge's theory of approximation and the complementary theory of global Cauchy in a few short steps. The other consequences of this binary theory are the theory of interpolation by Germay and the Metage-Leffler theory.

The approach through duplication is fully consistent with Kochi's approach to complex variables, because curved integrality is typical examples of linear functions. The basic condition of the book is a one-semester course in variables common to the university level, so that the elements of the local theory are assumed.

In particular, Cauchy's theory is assumed about the arena and the circle, but not the universal Kushite theory in any of its forms. The second author taught three times a graduate course based on this article at the University of Illinois, with good results..