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Complex Analysis in one Variable

  • Book
  • © 1985

Overview

Authors:
  1. Raghavan Narasimhan

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

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Elementary Theory of Holomorphic Functions

    • Raghavan Narasimhan
    Pages 1-52

  3. Covering Spaces and the Monodromy Theorem

    • Raghavan Narasimhan
    Pages 53-69

  4. The Winding Number and the Residue Theorem

    • Raghavan Narasimhan
    Pages 70-88

  5. Picard’s Theorem

    • Raghavan Narasimhan
    Pages 89-99

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

    • Raghavan Narasimhan
    Pages 100-118

  7. Applications of Runge’s Theorem

    • Raghavan Narasimhan
    Pages 119-143

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

    • Raghavan Narasimhan
    Pages 144-155

  9. Functions of Several Complex Variables

    • Raghavan Narasimhan
    Pages 156-165

  10. Compact Riemann Surfaces

    • Raghavan Narasimhan
    Pages 166-192

  11. The Corona Theorem

    • Raghavan Narasimhan
    Pages 193-215

  12. Subharmonic Functions and the Dirichlet Problem

    • Raghavan Narasimhan
    Pages 216-262

  13. Back Matter

    Pages 263-268
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Keywords

About this book

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.

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)

Authors and Affiliations

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

    Raghavan Narasimhan

Bibliographic Information

  • Book Title: Complex Analysis in one Variable

  • Authors: Raghavan Narasimhan

  • DOI: https://doi.org/10.1007/978-1-4757-1106-6

  • Publisher: Birkhäuser Boston, MA

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1985

  • eBook ISBN: 978-1-4757-1106-6Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: XVI, 268

  • Number of Illustrations: 3 b/w illustrations

  • Topics: Functions of a Complex Variable, Analysis, Applications of Mathematics, Topology

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