When the world was young and I was a student at UCLA I learned all sorts of folklore from my environment, particularly in the form of received wisdom from older students and even faculty. So it was that I came to learn that “God gave us analysis but the devil gave us topology” (this in connection with a profusion of Gegenbeispiele in the latter field that “doth make fools of us all,” at least as far as our intuition goes — just consider Alexander’s horned sphere). I learned that analytic life is best when the variable is complex, largely due to Cauchy’s integral formula (or, in truth, his integral theorem, which of course launches the whole business). I left school with such youthful prejudices in place, and started off lacking any real appreciation of either analysis or topology (I was nothing if not a contrarian). Nonetheless, as I have aged I have certainly come to appreciate analysis more and more, and I have in fact fallen deeply in love with topology — well, to be fair, with algebraic topology, where the destination of so many big functors is still a beloved category from algebra and some one like me is on familiar ground again. In any event, however, there is no question at all that my erstwhile seniors and professors were entirely on target in describing complex analysis as something of a mathematical paradise.
And now, after around thirty years of teaching across the undergraduate spectrum and having covered complex analysis quite a number of times, and in fact being in this particular saddle again this semester, I have also gained considerable familiarity with a variety of complex analysis texts, including Levinson-Redheffer’s Complex Analysis, Churchill’s Complex Variables and Applications, Howie’s Complex Analysis (which I reviewed in this column a few years ago), and even Ahlfors’ majestic Complex Analysis (used several years ago in a reading class for a particularly gifted student). Accordingly, I offer my comments about the book under present review from the viewpoint of its possible utility for such classroom activity, and my first observation is that the book reminds me a lot of nothing less than Levinson-Redheffer — which is of course quite a tribute.
What inspires this observation is that there is a substantial pedagogical similarity between these texts: both Levinson-Redheffer and Bruna-Cufi admit a two-fold purpose in that these books can be used to great effect as graduate as well as undergraduate text books. A propos, with the late Raymond Redheffer and the late Ernst Straus on the UCLA faculty when I was there, the former used his book in the undergraduate complex analysis course, whereas the later used this same book in the graduate course on complex analysis, which I in fact took: I found Straus’ presentation, using Levinson-Redheffer’s book quite intensely, remarkably lucid — far more so than was the case for my earlier undergraduate course in which another text was used by an instructor (not Redheffer) whose viewpoint was explicitly geometric (and which I have only come to appreciate with 20/20 hindsight). As far as this review goes, the point is that the way the book by Bruna and Cufi lends itself to the same sort of double dipping. The book is written in an accessible idiom and allows the instructor to pick and choose in such a manner as to build either a strong undergraduate course (chapters 1–5 plus a little of chapters 7 and 8) or a solid graduate course (add the rest of chapters 7 and 8, add chapters 9, 10, and 11, almost in their entirely). There is additional material to be had, too: chapter 6 deals with homology and introduces some de Rham theory, and the book finishes, in chapter 12, with some wonderful material on the complex Fourier transform.
This all makes for a very nice introduction to serious complex analysis, on the cusp of research level material, so to speak. The authors state, somewhat modestly, that they intend their book to serve as “a good complement to many of the references that are commonly used by both students and teachers.” I would state their case more strongly: this book should serve as a wonderful introduction to all the standard material for the PhD qualifying examination in complex analysis, provided the book is read carefully and covered completely, most if not all of the exercises are done. And then the diligent youngster will take the trouble to follow certain leads a little further, going to the indicated texts by, e.g., (yes, of course) Ahlfors, Hille, Nevanlinna-Paatero, Saks-Zygmund, and Rudin.
Thus, Bruna and Cufi have written an important and useful book, very readable, very elegant (including one of my favorite things, namely a ready derivation of Cauchy’s integral theorem from Green’s theorem), and pedagogically sound particularly as illustrated by their good choice of exercises. It will also serve well as a reference to professionals, particularly as regards what we do in the classroom, regardless of which principal texts we use in our complex analysis courses. Then again, there is no reason at all not to use the book under review itself: even its price is not out of reach.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.