Studies in Real and Complex Analysis

Prentice-Hall, 1965. - 219 pages. The eight articles in the present volume do not all presuppose the same mathematical background; they are directed generally to readers at the advanced undergraduate and first-year graduate level The initial article by H. J. Bremermann is a description of part of the modern theory of several complex variables which is centered about the successful efforts of mathematicians to understand fully the remarkable continuation properties possessed by analytic functions of several complex variables. Other topics central in this theory, such as the Cousin problems, analytic sets, etc. , are discussed, although more briefly. Graves' paper deals with a less extensive area, that of nonlinear functions from one Banach space to another, and in particular with the implicit function theorem. The material considered is treated in detail. Since this subject is beginning to make its way into advanced calculus texts, it is particularly fortunate to have this exposition. It is to be noted that Graves' paper has some elements in common with "Preliminaries to Functional Analysis" by Casper Goffman in Volume 1 of this series and that the two papers can be profitably read together. Hille's paper on semi-groups gives a brief description of this vast area of analysis. The reader is introduced to such central, structural features of semigroup theory as the resolvant and the infinitesimal generator, and is also afforded a hint of the applications of this theory to stochastic processes and partial differential equations, Hille's article also makes contact with that of Goffman referred to above. The article written by Hirschman and Widder is devoted to a relatively specific problem — the genesis of the real inversion formulas of the Laplace and Stieltjes transforms.