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© 1985

Complex Analysis in one Variable

Book

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Raghavan Narasimhan
    Pages 1-52
  3. Raghavan Narasimhan
    Pages 53-69
  4. Raghavan Narasimhan
    Pages 70-88
  5. Raghavan Narasimhan
    Pages 89-99
  6. Raghavan Narasimhan
    Pages 119-143
  7. Raghavan Narasimhan
    Pages 156-165
  8. Raghavan Narasimhan
    Pages 166-192
  9. Raghavan Narasimhan
    Pages 193-215
  10. Raghavan Narasimhan
    Pages 216-262
  11. Back Matter
    Pages 263-268

About this book

Introduction

This book is based on a first-year graduate course I gave three times at the University of Chicago. As it was addressed to graduate students who intended to specialize in mathematics, I tried to put the classical theory of functions of a complex variable in context, presenting proofs and points of view which relate the subject to other branches of mathematics. Complex analysis in one variable is ideally suited to this attempt. Of course, the branches of mathema­ tics one chooses, and the connections one makes, must depend on personal taste and knowledge. My own leaning towards several complex variables will be apparent, especially in the notes at the end of the different chapters. The first three chapters deal largely with classical material which is avai­ lable in the many books on the subject. I have tried to present this material as efficiently as I could, and, even here, to show the relationship with other branches of mathematics. Chapter 4 contains a proof of Picard's theorem; the method of proof I have chosen has far-reaching generalizations in several complex variables and in differential geometry. The next two chapters deal with the Runge approximation theorem and its many applications. The presentation here has been strongly influenced by work on several complex variables.

Keywords

Complex analysis Convexity Meromorphic function Monodromy Residue theorem Riemann surface corona theorem differential geometry function functional analysis holomorphic function integral integration residue sheaves

Authors and affiliations

  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

Bibliographic information

Reviews

"This book provides an alternative for a first-year graduate course in the classical theory of functions of one complex variable. A theme of the book is to relate classical complex analysis to other branches of mathematics. It includes many of the standard topics for a basic graduate course, but the exposition and viewpoint are strongly influenced by the theory of several complex variables.... One pleasant feature of the text is an early and elementary treatment of the theorems of Picard, Landau and Schottky via Ahlfors' extension of Schwarz's lemma in Chapter 4. In addition to covering many of the standard topics, the author also provides a treatment of covering spaces, the inhomogeneous Cauchy--Riemann equation, compact Riemann surfaces and Wolff's proof of the corona theorem."

--Mathematical Reviews (on the first edition)

"Provides a smooth and unintimidating transition from classical complex analysis in the plane to modern abstract theory on manifolds...An excellent, carefully written and thematically rich book which does not overwhelm the reader...Well-suited as a textbook either for sophisticated beginners or as a sequel to a one-semester introductory course."

---Jahresbericht der DMV (on the first edition)