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Principles of Complex Analysis

  • Textbook
  • © 2020

Overview

Authors:
  1. Serge Lvovski

    1. National Research University Higher School of Economics, Moscow, Russia, Federal Science Center System, Research Institute of Russian Academy of Sciences (FGU FNC NIISI RAN), Moscow, Russia

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  • Conformal mappings are introduced on an early stage, so the reader can learn to manipulate with subsets of the complex plane before passing to more sophisticated subjects
  • A special long section is devoted to evaluation of residues and evaluation of integrals using residues
  • The final chapter, which is devoted to Riemann surfaces, provides an elementary introduction into this subject which motivates the reader to study more technical parts of the theory

Part of the book series: Moscow Lectures (ML, volume 6)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Preliminaries

    • Serge Lvovski
    Pages 1-14

  3. Derivatives of Complex Functions

    • Serge Lvovski
    Pages 15-25

  4. A Tutorial on Conformal Maps

    • Serge Lvovski
    Pages 27-35

  5. Complex Integrals

    • Serge Lvovski
    Pages 37-48

  6. Cauchy’s Theorem and Its Corollaries

    • Serge Lvovski
    Pages 49-67

  7. Homotopy and Analytic Continuation

    • Serge Lvovski
    Pages 69-84

  8. Laurent Series and Isolated Singularities

    • Serge Lvovski
    Pages 85-97

  9. Residues

    • Serge Lvovski
    Pages 99-125

  10. Local Properties of Holomorphic Functions

    • Serge Lvovski
    Pages 127-138

  11. Conformal Maps. Part 1

    • Serge Lvovski
    Pages 139-162

  12. Infinite Sums and Products

    • Serge Lvovski
    Pages 163-188

  13. Conformal Maps. Part 2

    • Serge Lvovski
    Pages 189-210

  14. A Thing or Two About Riemann Surfaces

    • Serge Lvovski
    Pages 211-252

  15. Back Matter

    Pages 253-257
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Keywords

About this book

This is a brief textbook on complex analysis intended for the students of upper undergraduate or beginning graduate level. The author stresses the aspects of complex analysis that are most important for the student planning to study algebraic geometry and related topics. The exposition is rigorous but elementary: abstract notions are introduced only if they are really indispensable. This approach provides a motivation for the reader to digest more abstract definitions (e.g., those of sheaves or line bundles, which are not mentioned in the book) when he/she is ready for that level of abstraction indeed. In the chapter on Riemann surfaces, several key results on compact Riemann surfaces are stated and proved in the first nontrivial case, i.e. that of elliptic curves.


Reviews

“The book is well written, and the precision of the arguments given is carried out on a high level. This is undoubtedly a valuable book for students and can be recommended.” (Adam Lecko, Mathematical Reviews, April, 2022)

Authors and Affiliations

  • National Research University Higher School of Economics, Moscow, Russia, Federal Science Center System, Research Institute of Russian Academy of Sciences (FGU FNC NIISI RAN), Moscow, Russia

    Serge Lvovski

About the author

Serge Lvovski is associate professor at the Faculty of Mathematics of Higher School of Economics, Moscow, and research fellow in the Laboratory of Algebraic Geometry and its Applications.



Bibliographic Information

  • Book Title: Principles of Complex Analysis

  • Authors: Serge Lvovski

  • Series Title: Moscow Lectures

  • DOI: https://doi.org/10.1007/978-3-030-59365-0

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer Nature Switzerland AG 2020

  • Hardcover ISBN: 978-3-030-59364-3Published: 27 September 2020

  • Softcover ISBN: 978-3-030-59367-4Published: 28 September 2021

  • eBook ISBN: 978-3-030-59365-0Published: 26 September 2020

  • Series ISSN: 2522-0314

  • Series E-ISSN: 2522-0322

  • Edition Number: 1

  • Number of Pages: XIII, 257

  • Topics: Functions of a Complex Variable, Algebraic Geometry

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