Function theory of several variables
type: Item
instance of: scholarly article
title: Function theory of several variables
zbMATH Open document ID: 1391.32001
DOI: 10.1007/978-3-662-55542-2
author: Jörg Eschmeier
published in: Springer-Lehrbuch Masterclass
publication date: 2018-02-19T00:00:00Z
review text: The apprentice in the theory of functions of a single complex variable is inclined to explore on a naïve comparison basis several-variables analogues of the basic principles. The surprise is big, and fully justified, to discover a completely new landscape: the domain of convergence of a power series, analytic continuation, growth at infinity, integral representations, potential theory, orthogonal function expansions, classification of domains, groups of automorphisms are each revealing quite different phenomena. To learn and enter into this universe takes time, dedication, curiosity and especially a solid, authoritative guidance. The present book by Jörg Eschmeier offers to the German readership such a flawless, gentle, yet comprehensive, introduction to function theory of several complex variables. Compared to the several centuries old field of functions of one complex variable, the study of several complex variables is modern. It can be traced back to the pioneering works of Weierstrass and Poincaré, the foundational contributions of Hartogs, Oka, Weil, Siegel and the golden era, around mid XX-th century, marked by Cartan, Behnke, Stein, Serre, to name only a few prominent figures. The basic theory is nowadays well settled and should constitute, in harmony with the author's book clear intention, a part of the general mathematical curriculum of a Master Program in Mathematical Analysis. The research frontier in several complex variables remains active today in more sophisticated chapters contingent to algebraic geometry, mathematical physics, differential geometry of Cauchy-Riemann structures and global analysis. Each direction of current investigation would fill a separate volume and demand a rather long time of acquaintanceship. The book starts with the definition of holomorphy in several variables, elementary properties of analytic functions and a discussion of the possible convergence domains of a power series. Then emphasis is put on the structure of null sets…
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 32-01
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32C15
Mathematics Subject Classification ID: 32D05
zbMATH DE Number: 6840360
zbMATH Keywords: function theory of several complex variables
zbMATH Keywords: analytic continuation
zbMATH Keywords: domain of holomorphy
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: Cousin problems
zbMATH Keywords: Levi problem
zbMATH Keywords: Cauchy-Riemann equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-662-55542-2
OpenAlex ID: W2789516339
description: scientific article; zbMATH DE number 6840360