# Twenty-One Lectures on Complex Analysis

## A First Course

Textbook

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

1. Front Matter
Pages i-xii
2. Alexander Isaev
Pages 1-8
3. Alexander Isaev
Pages 9-16
4. Alexander Isaev
Pages 17-23
5. Alexander Isaev
Pages 25-32
6. Alexander Isaev
Pages 33-39
7. Alexander Isaev
Pages 41-48
8. Alexander Isaev
Pages 77-85
9. Alexander Isaev
Pages 167-173

### Introduction

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 complex analysis 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.

### Keywords

Complex Analysis Functions of Complex Variable Functional Analysis Homotopy Conformation transformations Cauchy's Independence of Homotopy Theorem Cauchy’s Integral Theorem Cauchy’s Integral Formula Fundamental Theorem of Algebra Riemann Mapping Theorem Mobius Transformations

#### Authors and affiliations

1. 1.Mathematical Sciences InstituteAustralian National UniversityActonAustralia

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
• Series Abbreviated Title SUMS
• DOI https://doi.org/10.1007/978-3-319-68170-2
• Copyright Information Springer International Publishing AG 2017
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-319-68169-6
• eBook ISBN 978-3-319-68170-2
• Series ISSN 1615-2085
• Series E-ISSN 2197-4144
• Edition Number 1
• Number of Pages XII, 194
• Number of Illustrations 30 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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