Complex Analysis: Theory and Applications

This is an introductory, but quite sophisticated, introduction to complex analysis. Although it starts with the definition of a complex number, it also uses terminology like “field” and “Abelian group” (without definition) as early as page 1, and matrices are used quite early on as well. The exposition is also fairly concise; in about 200 pages of text, the authors cover all the standard material in a first course on the subject (including some topics, like a proof of the Riemann mapping theorem, that are not always covered in such a course). The approach is modern: Homotopic paths are defined on page 51, for example, and a homotopic version of the Cauchy integral theorem is proved a bit more than 20 pages later. So, as these facts should make clear, this is not a book for neophytes.

 

There are six chapters of actual text. The first introduces complex numbers, the Riemann sphere, and stereographic projection. The second is on differentiability and holomorphic functions, including a discussion of differentiability in several contexts; a nice touch is the exploration of the relationship between complex differentiation and the concept of the derivative of a function between Euclidean spaces as a linear mapping. This chapter also introduces bilinear transformations and the various standard functions of complex variable theory (exponential, trigonometric, etc.).

 

Chapter 3 discusses integration, including, as previously noted, a fairly extensive treatment of homotopic paths and thus a fairly general version of the Cauchy integral theorem. The standard consequences of this theorem (e.g., Liouville’s theorem) are explored. Analytic branches of multivalued functions are also discussed in this chapter, but it should be noted that the authors, though talking about analytic branches of multi-valued functions, do stop short of mentioning the general concept of a Riemann surface.

 

Infinite series (Taylor and Laurent) are the subject of chapter 4, which also delves into the various kinds of points at which a function can fail to be analytic: poles, essential singularities, etc. Picard’s theorem is not mentioned, however. 

 

Chapter 5 is on residue theory and again covers the standard material on this subject: the residue theorem, argument principle, and Rouche’s theorem. There is a fairly extensive discussion of the use of this material to the evaluation of real integrals, and the Fundamental Theorem of Algebra, previously proved by the standard Liouville’s theorem argument, is proved again by using Rouche’s theorem.    

 

The final chapter of text is on conformal mappings, and culminates in both a statement and proof of the Riemann mapping theorem, proved here via Montel’s theorem, which is proved earlier in the chapter. 

 

Perhaps the most significant difference between this book and competing texts is the amount of space (chapter 7, comprising roughly half of the book’s 400 pages) that the authors devote to solutions to the exercises. Every exercise in the book, as far as I can tell, has a solution in these pages. Whether the easy availability of exercise solutions is a good thing is, of course, a matter of personal taste and perspective. These solutions certainly make this text extremely useful for a student engaged in self-study, particularly in preparation for something like a qualifying exam. Faculty members using a different text might well also appreciate having a convenient source of problems to punch up a lecture, assign as homework, or put on an exam. On the other hand, instructors who are using this book as a text, and who grade homework assignments, will have to look elsewhere when preparing assignments. 

 

Following this extensive solution set, there is a bibliography, but a number of the items listed here are in languages other than English: Romanian, Hungarian, French, and Russian. The Index that follows is fairly short (a bit more than two pages long) but does not, as far as I can tell, contain any glaring omissions. The book would, I think, have benefited from the inclusion of an index of notation.

 

The authors’ writing style is generally clear, albeit quite succinct. There is not much expository “filler” here, just a procession of theorems and proofs, punctuated occasionally by paragraphs labeled “Remarks”. This general lack of chattiness may prove daunting for undergraduates. The English is occasionally idiosyncratic, particularly in regards to the use of definite articles (e.g., one section is titled “Applications of the Residue Theorem to the Calculation of the Integrals”; another reads “Some consequences of Cauchy Formula”). Likewise, some terminology struck me as nonstandard; what I would call an annulus, for example, the authors refer to as an “open ring”.  However, none of this should cause a reader any serious difficulties. 

 

I do not know for sure, however, exactly what level of students this book is written for. The authors refer in the preface to the selection of topics presented here as being typical “for an undergraduate course”, yet the front cover of the book contains the word “graduate” at the top. The succinct exposition, level of sophistication of the material, and lack of conversational hand-holding probably make this book too difficult for most undergraduate students. On the other hand, as a text for a graduate course, the omission of topics such as Riemann surfaces and Picard’s Theorem may prove problematic. Perhaps, as alluded to earlier, the best use of the text would be as a source of self-study for graduate students preparing for qualifying exams.

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Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.