Visual Complex Functions

An Introduction with Phase Portraits

Authors: Wegert, Elias

  • First introduction to complex functions systematically using phase portraits (which are expected to become an indispensable tool for exploring complex functions in teaching, mathematical research and applications alike)
  • Reorganization of the traditional material covered by textbooks in complex analysis and emphasis on special aspects of the topic.
  • A companion to rather than a substitute for existing textbooks on complex analysis, although mostly self-contained
  • Enhances intuitive understanding of basic concepts and special functions by visualizing complex functions by phase portraits
  • Complete text equipped with color illustrations
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eBook $44.99
price for USA (gross)
  • ISBN 978-3-0348-0180-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.95
price for USA
  • ISBN 978-3-0348-0179-9
  • Free shipping for individuals worldwide
  • Online orders shipping within 2-3 days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
About this Textbook

This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – so-called phase portraits – which visualize functions as images on their domains.

Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is self-contained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes.

In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions.

Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.

About the authors

Elias Wegert is a Professor of Mathematics at the Technische Universität Bergakademie Freiberg. His main research interests are in Complex and Nonlinear Analysis; he is the author of a book on nonlinear Riemann-Hilbert problems and their applications. Wegert has experience in mathematical teaching at different levels for about 30 years. He is also committed to mathematical competitions for high school students.

Reviews

From the reviews:

“This textbook is an introduction to the classical theory of functions of one complex variable. Its distinctive feature are the graphical representations of functions, being the most useful tool in teaching and generally in mathematics. … The self-sufficiency of the textbook and the broad range of graphical examples makes the book useful for students as well as teachers of mathematics. … the book can be warmly recommended both to experts and to a new generation of mathematicians.” (Stanislawa Kanas, Zentralblatt MATH, Vol. 1264, 2013)

“Anyone who works with complex variables should read this book. … Visual Complex Functions is a beautiful and careful presentation of an entire advanced introduction to complex analysis based on phase portraits and, where appropriate, other kinds of computer-generated pictures. … My understanding of many ideas and phenomena deepened through reading this book.” (Lloyd N. Trefethen, SIAM Review, Vol. 55 (4), 2013)


Table of contents (7 chapters)

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-0348-0180-5
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $59.95
price for USA
  • ISBN 978-3-0348-0179-9
  • Free shipping for individuals worldwide
  • Online orders shipping within 2-3 days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Visual Complex Functions
Book Subtitle
An Introduction with Phase Portraits
Authors
Copyright
2012
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel
eBook ISBN
978-3-0348-0180-5
DOI
10.1007/978-3-0348-0180-5
Softcover ISBN
978-3-0348-0179-9
Edition Number
1
Number of Pages
XIV, 360
Number of Illustrations and Tables
29 b/w illustrations, 202 illustrations in colour
Topics