TwentyOne Lectures on Complex Analysis
A First Course
Authors: Isaev, Alexander
Free Preview 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 lecturebased teaching
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 About this Textbook

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 lecturebased 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 onesemester (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 onesemester undergraduate course.
 About the authors

Alexander Isaev is a professor of mathematics at the Australian National University. Professor Isaev’s research interests include several complex variables, CRgeometry, singularity theory, and invariant theory. His extensive list of publications includes three additional Springer books: Introduction to Mathematical Methods in Bioinformatics (ISBN: 9783540219736), Lectures on the Automorphism Groups of KobayashiHyberbolic Manifolds (ISBN: 9783540691518), and Spherical Tube Hypersurfaces (ISBN: 9783642197826).
 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)
 Table of contents (21 chapters)


Complex Numbers. The Fundamental Theorem of Algebra
Pages 18

ℝ and ℂDifferentiability
Pages 916

The Stereographic Projection. Conformal Maps. The Open Mapping Theorem
Pages 1723

Conformal Maps (Continued). Möbius Transformations
Pages 2532

Möbius Transformations (Continued). Generalised Circles. Symmetry
Pages 3339

Table of contents (21 chapters)
 Download Preface 1 PDF (1.2 MB)
 Download Sample pages 1 PDF (309 KB)
 Download Table of contents PDF (2.7 MB)
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Bibliographic Information
 Bibliographic Information

 Book Title
 TwentyOne Lectures on Complex Analysis
 Book Subtitle
 A First Course
 Authors

 Alexander Isaev
 Series Title
 Springer Undergraduate Mathematics Series
 Copyright
 2017
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer International Publishing AG
 eBook ISBN
 9783319681702
 DOI
 10.1007/9783319681702
 Softcover ISBN
 9783319681696
 Series ISSN
 16152085
 Edition Number
 1
 Number of Pages
 XII, 194
 Number of Illustrations
 30 b/w illustrations
 Topics