© 2012

Visual Complex Functions

An Introduction with Phase Portraits


Table of contents

  1. Front Matter
    Pages i-xiv
  2. Elias Wegert
    Pages 1-11
  3. Elias Wegert
    Pages 13-57
  4. Elias Wegert
    Pages 59-131
  5. Elias Wegert
    Pages 133-201
  6. Elias Wegert
    Pages 203-251
  7. Elias Wegert
    Pages 253-309
  8. Elias Wegert
    Pages 311-344
  9. Back Matter
    Pages 345-360

About this book


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.


complex functions phase portraits

Authors and affiliations

  1. 1., Institute of Applied AnalysisTU Bergakademie FreibergFreibergGermany

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.

Bibliographic information


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)