I must confess to a certain fondness for books of problems or counterexamples in mathematics. Unlike many other mathematics books, these do not have to be read linearly and don’t generally require large investments of time; you can open one to a page at random and happily kill a half hour or so thinking about a problem. And, of course, they often provide useful fodder for classes.

These considerations apply even more strongly when the subject matter of the text is a particularly interesting or elegant branch of mathematics, a description that I think many mathematicians would agree applies to the field of complex analysis. Yet, when I went looking for other problem books in complex function theory, I did not find a wealth of recent books to choose from. There is *Problems and Solutions in Complex Analysis* by Shakarchi (which is essentially a book-length solutions manual to Lang’s *Complex Analysis*), and another book with an interesting title that I have not seen called *Complex Variables: Principles and Problem Sessions* by Kapoor. There are also several books like *Berkeley Problems in Mathematics* by Paulo Ney de Souza and Jorge-Nuno Silva, and *Problems and Solutions in Mathematics* by Li et al., that reproduce questions (and provide solutions) from university qualifying exams and which include complex analysis among the subjects covered. Other than these I didn’t see very many books along these lines. So, given my interest in problem books in general and undergraduate complex analysis in particular, I was particularly looking forward to taking a look at Alpay’s book. It did not disappoint.

This is a substantial book (more than 500 pages long) which starts with a sketch of the construction of the field of complex numbers (defined both as a set of real matrices and also as the quotient space of the polynomial ring **R**[X] modulo the ideal generated by the polynomial X^{2} + 1) and proceeds to much more advanced material. The intended audience, according to the Preface, consists of undergraduates majoring in mathematics or electrical engineering, “with an eye on advanced students from both tracks.” I would often add the word “very” in front of “advanced”, particularly since there are some things discussed here (such as functional analysis) that are more usually addressed to graduate students — things like the Fock and Hardy spaces, for example, both of which are mentioned in chapter 5, were certainly not part of my undergraduate education, and another phrase (“reproducing kernel Hilbert space”), also used in chapter 5 and defined in chapter 14, is something I never heard of before looking at this book.

The book is in four parts, the first of which (which I will call the “pre-analytic” theory) consists of three chapters and discusses topics like the geometry of complex numbers, Möbius transformations, and sequences and series. Derivatives are mentioned, but only in the context of complex-valued functions of a real variable. Some of the exercises here are fairly standard (example: the cross ratio of four complex numbers is real if and only if they lie on a circle or line) and others are quite simple (describe geometrically the set of all complex numbers z which satisfy |z – *i*| = |z + *i*| ) but many are quite a bit more difficult. Purely as a matter of personal preference, I would liked to have seen more exercises relating complex numbers to the geometry of the plane (Needham’s *Visual Complex Analysis* has lots of them, without solutions) but of course not everybody shares my personal predilections.

The second part of the book (“Functions of a Complex Variable”), five chapters and about the half of the book in length, introduces differentiability and its consequences. Alpay initially draws a distinction between functions that are holomorphic in an open subset Ω of C (i.e., differentiable at every point in Ω) and those that are analytic (i.e., expressible as a power series at every point there) but of course the equivalence of these notions is discussed, and the other standard results that form the bulk of an undergraduate course (Cauchy’s integral theorem and formula, Laurent series, the theorems of Liouville, Morera, Schwartz and Rouché, evaluation of definite integrals by residue theory, etc.) are referred to and made the subject of a number of exercises.

I should mention that a student who does not already know what the Cauchy Integral formula is will have a hard time figuring it out from the section titled “Cauchy’s Formula and Applications”, since the formula that I refer to as the Cauchy integral formula, expressing f(a) in terms of the integral of f(z)/(z-a), does not explicitly appear in that section.

In view of the fact that the intended audience of this text consists not just of mathematics majors but electrical engineering majors as well, many of the exercises are of a rather applied nature, involving calculations with complicated integrals or series. Some of the solutions presented in this part of the book use results such as the Dominated Convergence Theorem or Monotone Convergence Theorem (topics that are covered in the final part of the book). This created a rather odd juxtaposition of difficulty levels within the text: surely a student with even passing existing familiarity with these results does not need to be explicitly reminded, for example, that a complex number is real if and only if it is equal to its conjugate, which is part of the content of Proposition 1.1.3 on page 12.

The final two parts of the book are considerably shorter than the first two. The third (“Applications and More Advanced Topics”) devotes one chapter each to harmonic functions, conformal mappings and linear system theory and signal processing; this last chapter is obviously a nod to the electrical engineering students who comprise part of the intended audience of this text. Finally, the fourth part of the book (“Appendix”) consists of various chapters summarizing, and giving exercises concerning, the results in real analysis, topology, functional analysis that are used in the book.

It should also be mentioned that this book is more than just a collection of exercises and solutions; there is some textual material as well, consisting of statements of theorems that are sometimes but not always accompanied by proofs. For example, the author spends almost four pages stating and proving a connection between absolutely convergent series and infinite products, and also devotes five pages to a direct topological proof of the Fundamental Theorem in Algebra. (The “usual” proof of this result, i.e., the one typically seen in a complex analysis course based on Liouville’s theorem, also appears here.) There is not, I think, enough textual material to use this book as a primary text for a course in complex analysis, but anyone teaching such a course would undoubtedly find this book a useful supplement, as would graduate students studying for their qualifier examinations, who could use this book to brush up on both theory and technique.

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

## Comments

## Henry Ricardo

This sounds like an interesting book. The first such book I found was

A Collection of Problems on Complex Analysisby Volkovyskii, Lunts, and Aramanovich (Pergamon Press, 1965). A more modern (and advanced) book isProblems in Real and Complex Analysisby Bernard R. Gelbaum (Springer, 1992).