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A First Course in Complex Analysis with Applications

Dennis G. Zill and Patrick D. Shanahan
Jones and Bartlett
Publication Date: 
Number of Pages: 
[Reviewed by
Donald L. Vestal
, on

As the title suggests, this is a book on complex analysis. Unlike the majority of complex analysis textbooks on the market, this book is directed at undergraduates. Any student with a competent background in calculus should find this text readable. And with a little supplementing, this text could also be used for a “cross-listed” course, i.e. a course taken by both undergraduates and graduate students.

The authors cover the standard material: the algebra of complex numbers, complex functions, mappings, analytic functions, the elementary functions, contour integrals, series and residues, and conformal mappings. One of the really nice things about the text is a comparison with real analysis at the end of some sections. For example, from the section on Cauchy’s Integral Formulas and Their Consequences, the authors point out that the differentiability of a complex function in some domain implies that all of the derivatives exist in that domain; by contrast there are real functions which are differentiable for all reals, but may not have a third (or higher) derivative at some points. These distinctions make for some good discussions (and a good review of real analysis).

The one thing I find lacking in this text is the notion of analytic continuation. Although all of the classic theorems are covered here (Cauchy-Goursat, the Cauchy Integral Formulas, Liouville’s Theorem, the Maximum Modulus Principle), there is no mention of analytic continuation and the notion of the uniqueness of analytic functions which take on certain values. Granted, the proof of such a result would be inappropriate for this text, but the principle is so important that it deserves at least some mention. (Even Picard’s Theorem gets a mention in one of the exercises.)

Although probably not suitable for a second semester of complex variables at the graduate level, this text is ideal for students who are being exposed to complex analysis for the first time. In particular, engineering students may find the applications at the end of each chapter very useful.

Donald L. Vestal is an Associate Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)


Chapter 1. Complex Numbers and the Complex Plane
                 1.1  Complex Numbers and Their Properties
                 1.2  The Complex Plane
                 1.3  Polar Form of Complex Numbers
                 1.4  Powers and Roots
                 1.5  Sets in the Complex Plane
                 1.6  Applications
                        Chapter 1 Review Quiz

Chapter 2. Complex Functions and Mapping
                 2.1  Complex Functions
                 2.2  Complex Functions as Mappings
                 2.3  Linear Mappings
                 2.4  Special Power Functions
                                  2.4.1  The Power Function zn
                                  2.4.2  The Power Function z1/n
                 2.5  Reciprocal Function
                 2.6  Limits and Continuity
                                   2.6.1  Limits
                                   2.6.2  Continuity
                 2.7  Applications
                       Chapter 2 Review Quiz

Chapter 3. Analytic Functions
                 3.1  Differentiability and Analyticity
                 3.2  Cauchy-Riemann Equations
                 3.3  Harmonic Functions
                 3.4  Applications
                       Chapter 3 Review Quiz
Chapter 4. Integration in the Complex Plane
                 4.1  Exponential and Logarithmic Functions
                                    4.1.1  Complex Exponential Function
                                    4.1.2  Complex Logarithmic Function
                 4.2  Complex Powers
                 4.3  Trigonmetric and Hyperbolic Functions
                                    4.3.1  Complex Trigonometric Functions
                                    4.3.2  Complex Hyperbolic Functions
                 4.4  Inverse Trigonometric and Hyperbolic Functions
                 4.5  Applications
                       Chapter 4 Review Quiz 

Chapter 5. Integration in the Complex Plane
                 5.1  Real Integrals
                 5.2  Complex Integrals
                 5.3  Cauchy-Goursat Theorem
                 5.4  Independence of Path
                 5.5  Cauchy's Integral Formulas and Their Consequences
                                   5.5.1  Cauchy's Two Integral Formulas
                                   5.5.2  Some Consequences of the Integral Formulas
                5.6  Applications
                      Chapter 5 Review Quiz 

Chapter 6. Series and Residues
                 6.1  Sequences and Series
                 6.2  Taylor Series
                 6.3  Laurent Series
                 6.4  Zeros and Poles
                 6.5  Residues and Residue Theorem
                 6.6  Some Consequences of the Residue Theorem
                                     6.6.1  Evaluation of Real Trigonometric Integrals
                                     6.6.2  Evaluation of Real Improper Integrals
                                     6.6.3  Integration Along a Branch Cut
                                     6.6.4  The Argument Principle
                                     6.6.5  Summing Infinite Series
                6.7  Applications
                      Chapter 6 Review Quiz

Chapter 7. Conformal Mappings
                 7.1  Conformal Mapping
                 7.2  Linear Fractional Transformations
                 7.3  Schwarz-Christoffel Transformations
                 7.4  Poisson Integral Formulas
                 7.5  Applications
                                    7.5.1  Boundary-Value Problems
                                    7.5.2  Fluid Flow
                       Chapter 7 Review Quiz
Appendixes:  I    Proof of Theorem 2.1
                    II   Proof of the Cauchy-Goursat Theorem
                   III  Table of Conformal Mappings

Answers for Selected Odd-Numbered Problems