The idea of complex numbers dates back at least 300 years—to Gauss and Euler, among others. Today complex analysis is a central part of modern analytical thinking. It is used in engineering, physics, mathematics, astrophysics, and many other fields. It provides powerful tools for doing mathematical analysis, and often yields pleasing and unanticipated answers.

This book makes the subject of complex analysis accessible to a broad audience. The complex numbers are a somewhat mysterious number system that seems to come out of the blue. It is important for students to see that this is really a very concrete set of objects that has very concrete and meaningful applications.


  • This new edition is a substantial rewrite, focusing on the accessibility, applied, and visual aspect of complex analysis
  • This book has an exceptionally large number of examples and a large number of figures.
  • The topic is presented as a natural outgrowth of the calculus. It is not a new language, or a new way of thinking.
  • Incisive applications appear throughout the book.
  • Partial differential equations are used as a unifying theme.

chapter Chapter 1|15 pages

Basic Ideas

chapter Chapter 2|10 pages

The Exponential and Applications

chapter Chapter 3|18 pages

Holomorphic and Harmonic Functions

chapter Chapter 4|30 pages

The Cauchy Theory

chapter Chapter 5|20 pages

Applications of the Cauchy Theory

chapter Chapter 6|13 pages

Isolated Singularities and Laurent Series

chapter Chapter 7|12 pages

Meromorphic Functions and Laurent Expansions

chapter Chapter 8|30 pages

The Calculus of Residues and Applications

chapter Chapter 9|18 pages

The Argument Principle

chapter Chapter 10|7 pages

The Maximum Principle

chapter Chapter 11|30 pages

The Geometric Theory of Holomorphic Functions

chapter Chapter 12|23 pages

Applications that Depend on Conformal Mapping

chapter Chapter 13|13 pages

Harmonic Functions

chapter Chapter 14|27 pages

The Fourier Theory

chapter Chapter 15|9 pages

Other Transforms