One Complex Variable from the Several Variable Point of View

by Peter V. Dovbush, Steven G. Krantz

Overview

Traditionally speaking, those who study the function theory of one complex variable spend little or no time thinking about several complex variables. Conversely, experts in the function theory of several complex variables do not consider one complex variable. One complex variable is the inspiration and testing ground for several complex variables, and several complex variables are the natural generalization of one complex variable.

The authors’ thesis here is that these two subject areas have much in common. These subject areas can gain a lot by learning to communicate with each other. These two fields are logically connected, and each can be used to explain and put the other into context. This is the purpose of this book.

The point of view and the methodology of the two subject areas are quite different. One complex variable is an aspect of traditional hard analysis. Several complex variables are more like algebraic geometry and differential equations, with some differential geometry thrown in. The authors intend to create a marriage of the function theory of one complex variable and the function theory of several complex variables, leading to a new and productive dialogue between the two disciplines.

The hope is for this book to foster and develop this miscegenation in a manner that leads to new collaborations and developments. There is much fertile ground here, and this book aims to breathe new life into it.

Product Details

ISBN-13:	9781041020141
Publisher:	CRC Press
Publication date:	06/29/2025
Series:	Textbooks in Mathematics
Pages:	410
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Peter V. Dovbush Dr. habil., Associate Professor, in Moldova State University, Institute of Mathematics and Computer Science.  He received his Ph.D. in Lomonosov Moscow State University in 1983 and Doctor of Sciences in 2003.  He has published over over 50 scholarly articles.

Steven G. Krantz is a Professor of Mathematics at Washington University in St. Louis.  He has previously taught at UCLA, Princeton University, and Penn State University.  He received his Ph.D. from Princeton University in 1974.  Krantz has directed 20 Ph.D. students and 8 Masters students. He has published over 130 books and over 300 scholarly articles.  He is the holder of the Chauvenet Prize and the Beckenbach Book Award and the Kemper Prize.  He is a Fellow of the American Mathematical Society.

Table of Contents

1      Introduction                                                                                                                  

2      Complex numbers                                                                                                     

3      The Cauchy-Riemann Equations                                                                     

4      Convergent Power Series

5      Montel’s Theorem And its Applications                                                 

6      Boundary Behavior of Holomorphic Functions                                  

7      Sokhotskii—Plemelj theorem

8      The Dirichlet Problem                                                                                     

9      The Ahlfors Map                                                                                                

10      Reproducing Kernels in Complex Analysis                                         

11      Conformal mappings                                                                                         

12      Fundamental Solutions in Complex Analysis: The Cauchy Riemann Operator