V. Karunakaran presents a distinctive introduction to the theory of a single complex variable in the second edition of Complex Analysis. The book has many noteworthy features. It contains, for example, a thorough list of alternate forms of the Cauchy-Riemann equations (section 1.9). Many readers will enjoy the rigorous treatment of Cauchy’s Theorem (chapter 4) and Cauchy’s Integral Formula. Instructors will see the Maximum Modulus Theorem proven via the Local Correspondence Theorem (a.k.a. the “m-fold covering lemma”) and the Open Mapping Theorem. Students may appreciate the solved exercises at the end of each chapter. The author clearly makes a consistent effort to highlight the topological aspects of introductory complex analysis as it flows through many of the topics listed above and continues on to the Riemann Mapping Theorem.
In the book’s introduction, the author indicates that the text was intentionally parsed into three semester-length parts: the first for undergraduates (Chapters 1 through 3), the second for introductory graduate students (Chapter 4), and the third for more advanced doctoral students (Chapter 5). Yet, despite the book’s thematic organization, it clearly only suits an introductory graduate course, being both too rigorous for an undergraduate and not comprehensive enough for advanced doctoral students. An undergraduate should not need a working knowledge of topology to appreciate complex variable theory and will, on the other hand, need an understanding of complex integration, which does not appear until chapter 4.
Perhaps the decision regarding what might be included in an advanced doctoral course in complex analysis is a matter of pedagogical preference. The reviewer would wish to see a better treatment of multivalent bounded analytic functions such as Blaschke products and related factorization theorems. From a pragmatic point of view, a reader might wish for a more comprehensive treatment of Laplace, Fourier and other transforms as well as a deeper discussion of the Cauchy, Poisson and other integral kernel.
Any reader of Complex Analysis — undergraduate, graduate, or researcher — will encounter a few stylistic oddities. Many sections of the first chapter (1.1–1.3, probably 1.5 and 1.7) can be omitted. The text presents theorems and proofs labeled mostly by section with occasional names included in parentheses. For this reason, a student first studying the subject may find it difficult to distinguish between important theorems and those which are mere stepping stones to greater heights. In addition, the book’s longer proofs are presented without any indication as to their general flow or method.
By all measures, a reader of Complex Analysis will receive a solid introduction to graduate complex analysis. Its treatment of topology will tie in nicely with other introductory courses. Proofs are complete and many difficult exercises are posed and fully solved. Anyone planning to teach a course in complex analysis should consider listing it as a reference.
For an undergraduate text, the reviewer would recommend instead Visual Complex Analysis by Needham. For introductory graduate studies, Conway’s Function of One Complex Variable or Rudin’s Real and Complex Analysis are better primary texts. When moving into advanced graduate work, Hoffman’s Banach Spaces of Analytic Functions, Garnett’s Bounded Analytic Functions, Duren’s Univalent Functions, or Burckel’s An Introduction to Classical Complex Analysis reflect the reviewer’s personal preferences.
Dov Chelst is the director of quantitative analysis at the ICMA Center for Public Safety Management. His main interests are mathematical optimization/programming, mathematical physics, and complex variables. He spends his spare time with his two daughters, and occasionally attends a Tae Kwon Do class.
A Bird's Eye View of the Complex Plane;
Elementary Properties of Analytic Functions;
Complex Integral Calculus;
Riemann Mapping Theorem