Introduction to Complex Analysis

This is an interesting, though rather demanding, text for an introductory graduate course in complex analysis. There is enough material here for a year-long course, but the book can easily be adapted to a course lasting only one semester. It is characterized by a strong emphasis on real variable techniques and applications, as well as by an occasionally nonstandard point of view and selection of topics. All of the “usual suspects” of an introductory complex analysis course are covered here, but the book does not stop there, and also includes material that is not generally found in books at this level, some of which (if my own experience is to be considered representative) may not be well known even to non-specialist faculty members. 

 

Even though this book appears in the Graduate Studies in Mathematics series, an advertising blurb on the AMS webpage for this book states that it could be used for “well-prepared undergraduates”. If so, these undergraduates would have to be more “well-prepared” than any that I have previously encountered at Iowa State University. For reasons that I hope will be made clear in what follows, this is very much a graduate-level book, and reading it requires a considerable level of mathematical maturity. 

 

The text begins, as do most complex analysis texts, with some introductory material on complex numbers, but the author doesn’t tarry here and proceeds quite quickly to calculus-related matters, such as infinite sequences and series. Holomorphic functions are defined and related to the notion of differentiability of functions from the Euclidean plane (or appropriate subsets thereof) to itself. The connection between holomorphic functions and those representable by power series is also explored; it is proved that convergent power series define holomorphic functions. The basic functions of complex analysis (trigonometric, exponential, logarithm) are defined for real numbers via reference to the differential equations they satisfy and then extended to complex arguments. In the concluding section of this chapter a proof is given that \( \pi^{2} \) (and therefore, of course, \( \pi \)) is irrational. 

 

Chapter 2 covers many of the standard topics of a complex analysis course, specifically the Cauchy integral theorem and formula, and consequences thereof: Liouville’s theorem, the Fundamental Theorem of Algebra (two proofs of which are given, one avoiding Liouville’s theorem), the maximum principle, Schwarz reflection principle, analytic continuation,  infinite products, singularities, Taylor and Laurent series, etc. This chapter completes the proof of the equivalence, started in the first chapter, between holomorphic functions and those representable by Taylor series.

 

From Laurent series, we move in chapter 3 to Fourier series, and from there to Fourier transforms and Laplace transforms, which are related to notions in complex analysis. An appendix to this chapter proves and uses the Stone Weierstrass theorem. Chapter 4 then starts with residue calculus and its application to the evaluation of definite integrals and moves to a study of both the Gamma function and the Riemann zeta function.

 

Chapter 5 looks at conformal mappings and Riemann surfaces. Among the “big” theorems proved here are the Riemann Mapping Theorem and Picard’s theorem. Some nonstandard appendices to this chapter explore topics related to topology (covering maps) and differential geometry (including non-Euclidean spaces like the Poincare disc and its associated metric tensor). 

 

Riemann surfaces show up again in the next chapter, which discusses elliptic functions and elliptic integrals. Then, in the seventh and final chapter, the relationship between differential equations and complex analysis is discussed, first discussing Bessel’s equation and then discussing the more general theory of linear differential equations with coefficients that are holomorphic in some domain. Differential equations, as with Fourier series and integrals from chapter 3, are discussed in considerably more depth than is typical for textbooks at this level. 

 

After the main text material, there are a number of Appendix sections. Some discuss various topics in advanced calculus and analysis such as metric spaces, differentiation between multi-dimensional Euclidean spaces (including the inverse function theorem, proved via the contraction mapping principle), Tauberian theorems and other topics. One particularly unusual one discusses the use of complex analysis to solve polynomial equations of low degree, including the use of a special function called the Bring radical (introduced in chapter 7), to address quintic polynomials. 

 

I suspect that the Appendix sections on advanced calculus will not prove an entirely adequate substitute for some actual prior exposure to these topics, so a good real analysis background should be seen as a prerequisite for reading this text.  On occasion, the Lebesgue integral is referred to, but a student without a prior course in measure-theoretic real analysis will not, I think, find this a serious impediment. Some prior background in linear algebra is also useful. In one amusing sentence, the author begins a discussion involving linear algebra by saying that the prior background can be obtained in reference [49]; this turns out to be a book by the author on the subject, described in this reference as “Preprint” with no link as to where to find it. However, a few seconds on google reveals the location of the the book draft on the author’s website.  According to the AMS web page, this book will be published in September 2020.

 

So, to summarize and conclude: there is a lot of very interesting information in this book, and a graduate course based on it should be an informative one. However, as I said in the first sentence of this review, this book is also a demanding one, and some—perhaps many—instructors may see this text as too difficult for the graduate courses that they teach. A student will have to work hard as he or she progresses through it, but strong students should find that the net result of doing so—a good, solid understanding of the modern theory of complex analysis, and a real sense of how professional analysts ply their trade—is worth the effort.

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