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Complex Analysis in One Variable

  • Textbook
  • © 2001

Overview

Authors:
  1. Raghavan Narasimhan

    1. Department of Mathematics, University of Chicago, Chicago, USA

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  2. Yves Nievergelt

    1. Department of Mathematics, Eastern Washington University, Cheney, USA

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Table of contents (24 chapters)

  1. Front Matter

    Pages i-xiv
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  2. Complex Analysis in One Variable

    1. Front Matter

      Pages 1-1
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    2. Elementary Theory of Holomorphic Functions

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 3-51

    3. Covering Spaces and the Monodromy Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 53-68

    4. The Winding Number and the Residue Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 69-85

    5. Picard’s Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 87-96

    6. The Inhomogeneous Cauchy-Riemann Equation and Runge’s Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 97-114

    7. Applications of Runge’s Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 115-137

    8. The Riemann Mapping Theorem and Simple Connectedness in the Plane

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 139-149

    9. Functions of Several Complex Variables

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 151-160

    10. Compact Riemann Surfaces

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 161-185

    11. The Corona Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 187-208

    12. Subharmonic Functions and the Dirichlet Problem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 209-252

  3. Back Matter

    Pages 253-253
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  4. Exercises

    1. Front Matter

      Pages 255-255
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    2. Introduction

      • Yves Nievergelt
      Pages 257-257

    3. Review of Complex Numbers

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 259-266

    4. Elementary Theory of Holomorphic Functions

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 267-295

    5. Covering Spaces and the Monodromy Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 297-304

    6. The Winding Number and the Residue Theorem

      • Raghavan Narasimhan, Yves Nievergelt
      Pages 305-312
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Keywords

About this book

The original edition of this book has been out of print for some years. The appear­ ance of the present second edition owes much to the initiative of Yves Nievergelt at Eastern Washington University, and the support of Ann Kostant, Mathematics Editor at Birkhauser. Since the book was first published, several people have remarked on the absence of exercises and expressed the opinion that the book would have been more useful had exercises been included. In 1997, Yves Nievergelt informed me that, for a decade, he had regularly taught a course at Eastern Washington based on the book, and that he had systematically compiled exercises for his course. He kindly put his work at my disposal. Thus, the present edition appears in two parts. The first is essentially just a reprint of the original edition. I have corrected the misprints of which I have become aware (including those pointed out to me by others), and have made a small number of other minor changes.

Reviews

"The first part of the book under review represents essentially the material of R. Narasimhan’s 'Complex analysis in one variable' (first edition, 1985). The second part of the book, authored by Y. Nievergelt, consists of exercises and relevant references…. There are notes at the end of each chapter which contain brief remarks on the history of the material presented as well as references to the literature. The exercises of part II give the reader the opportunity to consolidate his knowledge in complex analysis. At the end of this part there are notes for the exercises and references. The book can be highly recommended for a thorough study of complex analysis."

—ZENTRALBLATT MATH (Review of the second edition)

"The book introduces and makes use of concepts from many different areas of mathematics, especially ideas used in several complex variables and differential geometry. There is also a short…introductory chapter dealing with several complex variables…. The exercises in Part 2 vary from basic to advanced, and provide good practice for the concepts and techniques of the subject…. The choice of topics covered gives an excellent introduction to modern complex analysis. The exposition is well written. All in all, this book is a welcome addition to the list of books presenting a first course in complex analysis."

—MATHEMATICAL REVIEWS (Review of the second 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 (Review of the first edition)

Authors and Affiliations

  • Department of Mathematics, University of Chicago, Chicago, USA

    Raghavan Narasimhan

  • Department of Mathematics, Eastern Washington University, Cheney, USA

    Yves Nievergelt

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