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Complex Variables and Analytic Functions: An Illustrated Introduction

Bengt Fornberg, Cécile Piret
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on
This new undergraduate-level textbook on complex analysis has, as its principal audience, students who wish to ultimately use complex variables in applications, rather than to develop a theoretical appreciation of the subject. This applications-oriented approach (which of course is not a surprising one, given the publisher of the text) manifests itself in several ways, primarily the choice of topics covered and the method of presentation. 
With regard to the former, the book covers the standard content of a one-semester course, including topics like the basic properties of complex numbers and analytic functions, the Cauchy integral theorem and integral formula, Liouville’s theorem, the fundamental theorem of algebra, Taylor and Laurent series, singularities, residue theory, bilinear transformations and conformal mappings, and analytic continuation. The text also includes topics that are useful for budding applied mathematicians and engineers (including the Fourier, Laplace, Mellin and Hilbert transforms; special functions and ODEs, including an extended discussion of the Painlevé equation; and integral approximation). The text focuses on mastery of the ideas that are used in applications rather than providing a specific discussion of the applications themselves.  
More sophisticated theoretical topics (such as for example, the homotopic version of the Cauchy integral theorem) are not covered, but at the same time, some topics (like analytic continuation) are covered in somewhat more depth than is typical in books at this level. (The authors give several points of view for this topic, pointing out that the traditional circle-chain method “is quite impractical in many contexts.”) There is enough material in the text for a year’s course, but it can be readily adapted to one-semester courses as well; the first five chapters cover most of the “standard content” discussed above, and the early parts of chapter 8 discuss conformal mapping and bilinear transformations. 
The method of presentation of the material is also geared towards applied mathematicians. There are certainly plenty of proofs, but the authors do not hesitate to replace a proof by a heuristic argument if circumstances warrant. They do so in an honest way, however, and do not attempt to pass off a heuristic discussion as a proof. As an example, the authors give what they themselves refer to as a heuristic motivation for the Riemann mapping theorem, the culmination of which is the deduction of the existence of a function satisfying certain properties by reference to the physical existence of an electrostatic potential in a certain configuration.
In a number of cases, results are stated without proof in the main body of the text and the proof is deferred until the end of a chapter, in a section marked “Supplemental Materials”, so as to be easily omitted if the instructor wishes to do so. Some end-of-chapter exercises call for proofs, but most are of a computational nature. The preface suggests that a solutions manual may be forthcoming, but as of this writing, I cannot find one on the publisher’s webpage.
Another interesting and significant feature of the method of presentation, prominently reflected in the title of the book, is the use of illustrations. Almost 25 years ago, Tristan Needham wrote Visual Complex Analysis, a book that, among other things, approached this subject from a highly visual point of view. Needham’s book is fun to read and contains a lot of interesting material, but its idiosyncratic style (it doesn’t discuss the derivative until almost 200 pages into the text, for example, and then does so in a nonstandard way) makes it a difficult book to use as a text. This book also makes use of illustrations and visualization (many of the pictures in the text are in full color) but, being more standard in presentation, is more suitable for use as a textbook. The illustrations serve a number of purposes: they illustrate Riemann surfaces (here called “Riemann sheets”), for example, and also help visualize complex-valued functions of a complex variable by showing graphs of their real and imaginary parts.
There is a good bibliography at the end of the text, consisting largely of books but also containing some references to research papers. In an unusual and welcome feature, for each bibliographic entry, there is a reference to the page or pages of the text on which that reference is cited.
Finally, I cannot end this review without mentioning one interesting non-mathematical fact that I learned from this book: Paul Painlevé, for whom the Painlevé equation is named, was not only a mathematician but also the Prime Minister of France. 
To summarize and conclude: if you’re teaching a course in complex analysis with an applied slant, or if you are an applied mathematician, engineer or physicist who needs a good reference for complex analysis, this book merits a serious look.


Mark Hunacek ( teaches mathematics at Iowa State University. 
The table of contents is not available.