Every number theorist has a soft spot in his heart for the prime number theorem, a.k.a. PNT. I have nothing but idyllic memories of the late E. G. Straus’ series of lectures on the elementary Selberg-Erdös proof and still have my 35 year old notes sequestered in a safe place in one of my desk drawers. In the 1990s I got hold of the compact (approx. 80 pp.) Springer GTM volume, *Analytic Number Theory*, by Donald J. Newman (now also deceased), containing his own proof of PNT: a truly amazing and beautiful piece of work. And the same can be said about the rest of that book, particularly Newman’s discussion of Waring’s conjecture, presumably patterned after his 1960 paper, “A simplified proof of Waring’s conjecture.”

The book under review, innocently titled, *Complex Analysis*, in fact also features PNT *à la* Newman, and we read in the Preface to the [present] Third Edition that

the most significant changes in this edition revolve around the proof of the prime number theorem. There are two new sections… on Dirichlet series [yielding that] a pivotal result on the zeta function, which seemed to “come out of the blue,” is now seen in the context of the analytic continuation of Dirichlet series. Finally the actual proof of the prime number theorem has been considerably revised. The original proofs by Hadamard and de la Valée Poussin were both long and intricate … Newman’s 1980 article presented a dramatically simplified approach. Still the proof relied on several nontrivial … results … due to Chebychev, which formed a separate appendix in the earlier editions … [In the present edition we] followed [a refining approach by Zagier, and now] the proof relies … on only one relatively straightforward result due [to] Chebychev.

This, by itself, is worth the price of admission — at least for us number theorists.

But there’s a lot more. *Complex Analysis* will also serve as a fine textbook for a general first course in complex variables, particularly given that the authors’ goals include (cf. the Preface to the Second Edition) “present[ing] the theory of analytic functions with as little dependence as possible on advanced concepts from topology and several-variable calculus … to highlight the authentic complex-variable methods and arguments as opposed to those of other mathematical areas.” I believe that this is a very sound pedagogical move: having suffered though an earlier heavily geometric topological treatment of the subject in my undergraduate days I was fortunate enough to get a graduate level course on the same material from the same Straus who had introduced me to PNT some years before, and I was immediately converted to his way of doing complex analysis, especially Cauchy theory. I wonder, perhaps a bit parochially, if the fact that both he and Newman were hard-core-users of complex analytic methods were responsible for the according “classical” treatment of the according material. In any case it is my favorite, and I think the best, way to teach the subject at the undergraduate level, because it is obviously the most accessible: it requires the least amount of preparation.

Thus, Bak-Newman present a very readable and accessible treatment of the subject, covering enough for a year long course at the undergraduate level (go with the first part of the book for a semester course), as well as opportunities for a special seminar or something akin thereto. In other words, it’s all there, and more, starting with analyticity (including some novelties, e.g. “analytic polynomials”), covering Cauchy theory very nicely (with a proper treatment of power series woven throughout), and covering a host of deep and useful material in the later chapters. *Complex Analysis* gives plenty of coverage, for example, to the use of the residue calculus to evaluate improper real integrals (and even some proper but nasty ones), including a discussion of shifting contours; the discussion of conformal mapping flows into coverage of the Riemann mapping theorem; Phragmén and Lindelöf appear right after the maximum modulus principle; and infinite products appear toward the end of the book, as do the gamma ands zeta functions. And, to be sure, the very last section, “Applications to other areas of mathematics,” includes both the Fourier uniqueness theorem and, as already mentioned, PNT. Both of these themes would be fine choices for an advanced undergraduate seminar.

Indeed, if a student were to work through every page of this book with care and persistence, paying proper attention to the copious accessible and well thought out exercises that it comes with, he would find himself well ahead of the game in graduate school.

There is no doubt about it: Bak-Newman’s *Complex Analysis* is a wonderful book.

See also our review of the second edition.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.