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Twenty-One Lectures on Complex Analysis

A First Course

  • Textbook
  • © 2017

Overview

Authors:
  1. Alexander Isaev

    1. Mathematical Sciences Institute, Australian National University, Acton, Australia

    View author publications

    You can also search for this author in PubMed   Google Scholar

  • Clear and rigorous exposition is supported by engaging examples and exercises
  • Provides a means to learn complex analysis as well as subtle introduction to careful mathematical reasoning
  • Topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Undergraduate Mathematics Series (SUMS)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Complex Numbers. The Fundamental Theorem of Algebra

    • Alexander Isaev
    Pages 1-8

  3. ℝ- and ℂ-Differentiability

    • Alexander Isaev
    Pages 9-16

  4. The Stereographic Projection. Conformal Maps. The Open Mapping Theorem

    • Alexander Isaev
    Pages 17-23

  5. Conformal Maps (Continued). Möbius Transformations

    • Alexander Isaev
    Pages 25-32

  6. Möbius Transformations (Continued). Generalised Circles. Symmetry

    • Alexander Isaev
    Pages 33-39

  7. Domains Bounded by Pairs of Generalised Circles. Integration

    • Alexander Isaev
    Pages 41-48

  8. Primitives Along Paths. Holomorphic Primitives. The Existence of a Holomorphic Primitive of a Function Holomorphic on a Disk. Goursat’s Lemma

    • Alexander Isaev
    Pages 49-55

  9. Proof of Lemma 7.2. Constructible Primitives of Holomorphic Functions along Paths. Integration of Holomorphic Functions over Arbitrary Paths. Homotopy. Simply-Connected Domains. The Riemann Mapping Theorem

    • Alexander Isaev
    Pages 57-66

  10. Cauchy’s Independence of Homotopy Theorem. Integration over Piecewise C 1-paths. Jordan Domains and Integration over their Boundaries

    • Alexander Isaev
    Pages 67-76

  11. Cauchy’s Integral Theorem. Proof of Theorem 3.1. Cauchy’s Integral Formula

    • Alexander Isaev
    Pages 77-85

  12. Morera’s Theorem. Sequences and Series of Functions. Uniform Convergence Inside a Domain. Power Series. Abel’s Theorem. Disk of Convergence. Radius of Convergence

    • Alexander Isaev
    Pages 87-95

  13. Proof of Theorem 11.9. Power Series (Continued). Taylor Series. Local Power Series Expansion of a Holomorphic Function. Cauchy’s Inequalities. The Uniqueness Theorem

    • Alexander Isaev
    Pages 97-105

  14. Liouville’s Theorem. Laurent Series. Annulus of Convergence. Laurent Series Expansion of a Function Holomorphic on an Annulus. Cauchy’s Inequalities. Isolated Singularities of Holomorphic Functions

    • Alexander Isaev
    Pages 107-116

  15. Isolated Singularities of Holomorphic Functions (Continued). Characterisation of an Isolated Singularity via the Laurent Series Expansion. Orders of Poles and Zeroes. Casorati-Weierstrass’ Theorem. Isolated Singularities of Holomorphic Functions at ∞ and their Characterisation via Laurent Series Expansions

    • Alexander Isaev
    Pages 117-125

  16. Isolated Singularities of Holomorphic Functions at ∞ (Continued). Orders of Poles at ∞. Casorati-Weierstrass’ Theorem for an Isolated Singularity at ∞. Residues. Cauchy’s Residue Theorem. Computing Residues

    • Alexander Isaev
    Pages 127-135

  17. Computing Residues (Continued). Computing Integrals over the Real Line Using Contour Integration. The Argument Principle

    • Alexander Isaev
    Pages 137-145

  18. Index of a Path. The Argument Principle (Continued). Rouché’s Theorem. Theorem 1.1 Revisited. Proof of Theorem 3.2. The Maximum Modulus Principle. Proof of Theorem 3.3

    • Alexander Isaev
    Pages 147-155

  19. Schwarz’s Lemma. Conformal Maps of the Unit Disk and the Upper Half-Plane. (Pre)-Compact Subsets of a Metric Space. Continuous Linear Functionals on H(D). Arzelà-Ascoli’s Theorem. Montel’s Theorem. Hurwitz’s Theorem

    • Alexander Isaev
    Pages 157-165

  20. Analytic Continuation

    • Alexander Isaev
    Pages 167-173
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Keywords

About this book

At its core, this concise textbook presents standard material for a first course in complex analysis at the advanced undergraduate level. This distinctive text will prove most rewarding for students who have a genuine passion for mathematics as well as certain mathematical maturity. Primarily aimed at undergraduates with working knowledge of real analysis and metric spaces, this book can also be used to instruct a graduate course. The text uses a conversational style with topics purposefully apportioned into 21 lectures, providing a suitable format for either independent study or lecture-based teaching. Instructors are invited to rearrange the order of topics according to their own vision. A clear and rigorous exposition is supported by engaging examples and exercises unique to each lecture; a large number of exercises contain useful calculation problems. Hints are given for a selection of the more difficult exercises. This text furnishes the reader with a means of learning complexanalysis as well as a subtle introduction to careful mathematical reasoning. To guarantee a student’s progression, more advanced topics are spread out over several lectures.

 

This text is based on a one-semester (12 week) undergraduate course in complex analysis that the author has taught at the Australian National University for over twenty years. Most of the principal facts are deduced from Cauchy’s Independence of Homotopy Theorem allowing us to obtain a clean derivation of Cauchy’s Integral Theorem and Cauchy’s Integral Formula.  Setting the tone for the entire book, the material begins with a proof of the Fundamental Theorem of Algebra to demonstrate the power of complex numbers and concludes with a proof of another major milestone, the Riemann Mapping Theorem, which is rarely part of a one-semester undergraduate course.

Reviews

“This text furnishes the reader with a means of learning complex analysis as well as a subtle introduction to careful mathematical reasoning. …There is no doubt that graduate students and seasoned analysts alike will find a wealth of material in this project and appreciate its particular construction.” (Vicenţiu D. Rădulescu, zbMATH 1386.30001, 2018)

Authors and Affiliations

  • Mathematical Sciences Institute, Australian National University, Acton, Australia

    Alexander Isaev

About the author

Alexander Isaev is a professor of mathematics at the Australian National University. Professor Isaev’s research interests include several complex variables, CR-geometry, singularity theory, and invariant theory. His extensive list of publications includes three additional Springer books: Introduction to Mathematical Methods in Bioinformatics (ISBN: 978-3-540-21973-6), Lectures on the Automorphism Groups of Kobayashi-Hyberbolic Manifolds (ISBN: 978-3-540-69151-8), and Spherical Tube Hypersurfaces (ISBN: 978-3-642-19782-6).

Bibliographic Information

  • Book Title: Twenty-One Lectures on Complex Analysis

  • Book Subtitle: A First Course

  • Authors: Alexander Isaev

  • Series Title: Springer Undergraduate Mathematics Series

  • DOI: https://doi.org/10.1007/978-3-319-68170-2

  • Publisher: Springer Cham

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

  • Copyright Information: Springer International Publishing AG 2017

  • Softcover ISBN: 978-3-319-68169-6Published: 07 December 2017

  • eBook ISBN: 978-3-319-68170-2Published: 29 November 2017

  • Series ISSN: 1615-2085

  • Series E-ISSN: 2197-4144

  • Edition Number: 1

  • Number of Pages: XII, 194

  • Number of Illustrations: 30 b/w illustrations

  • Topics: Analysis

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