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Springer Undergraduate Mathematics Series

Twenty-One Lectures on Complex Analysis

A First Course

Authors: Isaev, Alexander

  • 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
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Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-3-319-68170-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA
  • ISBN 978-3-319-68169-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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 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.

About the authors

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).

Table of contents (21 chapters)

  • Complex Numbers. The Fundamental Theorem of Algebra

    Isaev, Alexander

    Pages 1-8

  • ℝ- and ℂ-Differentiability

    Isaev, Alexander

    Pages 9-16

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

    Isaev, Alexander

    Pages 17-23

  • Conformal Maps (Continued). Möbius Transformations

    Isaev, Alexander

    Pages 25-32

  • Möbius Transformations (Continued). Generalised Circles. Symmetry

    Isaev, Alexander

    Pages 33-39

Buy this book

eBook $39.99
price for USA (gross)
  • ISBN 978-3-319-68170-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $49.99
price for USA
  • ISBN 978-3-319-68169-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Twenty-One Lectures on Complex Analysis
Book Subtitle
A First Course
Authors
Series Title
Springer Undergraduate Mathematics Series
Copyright
2017
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing AG
eBook ISBN
978-3-319-68170-2
DOI
10.1007/978-3-319-68170-2
Softcover ISBN
978-3-319-68169-6
Series ISSN
1615-2085
Edition Number
1
Number of Pages
XII, 194
Number of Illustrations and Tables
30 b/w illustrations
Topics