A Quick Introduction to Complex Analysis

My department is in a state of (d)evolution: it’s growing by leaps and bounds. Keen-eyed youths with PhDs in different parts of applied mathematics are sprouting up in large numbers and with increasing frequency in response to a clear supply-and-demand dynamic, and we are faced with a sort of perestroika — I’m not so sure about any purported glasnost, however. As a dinosaur, I am dismayed by the mounting pressure, coming in waves from beyond the confines of my campus, to teach every kid everything from Maple to Mathematica, with diminishing regard for what babies are thrown out with this high-tech (high-TeX?) bathwater. I am pretty sure that this description applies across the fruited plain and likely well beyond, given the influence we wield.

Getting back to local data, i.e. my department, with the foregoing parameters in place we are now faced with a nasty corollary of the pigeonhole principle: with only a certain number of schedule spots available and more “relevant” courses elbowing their way into the regular line-up, complex analysis has now devolved into an elective without a prerequisite course in how to do proofs. The latter sacrifice happened long ago, and now it’s gotten even worse: Johnny and Jane can get their undergraduate mathematics degrees without knowing any Cauchy theory and without ever having seen a Riemann surface. But they all can program up a storm, of course.

So, all right, that’s the nature of the modern beast, then, and I guess I have lamented this state of affairs long enough: the juggernaut has been unleashed. But, to be sure, even this dark a cloud has its silver lining: with fewer and fewer of my colleagues inclined in this direction, I at least get to teach complex analysis with great frequency — and do manage to throw in a decent amount of serious mathematics: proofs, yes, even if not to the same extent as in the good old days. And it is fun to compute all those improper real integrals by exploiting the residue calculus.

Under this new regime, a title like that of the book under review is immediately eye-catching: it looks like a good candidate for the kind of course described above, certainly in contrast to, say, Ahlfors’ classic book (which, admittedly, was marketed as a first-year graduate text even back in 1966; still, I can dream, can’t I?). The back-cover of the present book states that “[t]he aim of this book is to give a smooth analytic continuation from calculus to complex analysis by way of plenty of practical examples and worked-out exercises.” So, what does this mean? It turns out that it means a lot: I think the book is pretty much on target (again, given the parameters outlined above), with a major caveat in place, which I’ll elaborate momentarily. In other words, it’s one of those “there’s good news, and there’s bad news” affairs.

Let’s take the central theme of Cauchy theory as a case in point. The bad news is that the centerpiece of the entire subject, the Cauchy Integral Theorem, is presented on p. 3 with the comment that it is “the most fundamental [theorem] in complex analysis and one of the most far-reaching theorem[s] in [all of] mathematics. “ Yes, truer words were never spoken, but where’s the proof? Well, it occurs in the second part of the book, on p. 125, after coverage of Green’s Theorem. And I think that’s bad news indeed. On the other hand, right after presenting this titanic result as a black box, the authors go on to reap the various rich dividends, reaching residue theory as early as p. 20 and the Cauchy Integral Formula on p. 27. And happily these results are provided with the required proofs, even if on occasion the authors get a little sketchy: see, e.g., their proof of the residue theorem on p. 21 (it’s all there, though).

Beyond this, very reasonably, and very properly, in this first part of the book the authors build heavily on the subject of power series, with which their charges are purportedly already familiar, what with calculus supposedly recently covered. Thus, with all this in place, we get some real gems early on: \(\displaystyle\int_0^\infty \frac{\sin(x)}{x}\,dx = \frac{\pi}{2}\) on p. 23; Bernoulli and Euler numbers on pp. 24–25 (without complex analysis, actually); Liouville’s Theorem on p. 28 (with a mention of the Fundamental Theorem of Algebra — I would have like to have seen the proof); \[\sin(z)=z\prod_{n\in\mathbb{Z}}\left(1-\frac{z}{\pi n}\right)\exp\left(\frac{z}{\pi n}\right)\] on p. 43 (in the context of flows around vortices: an interesting approach); and, yes indeed, the product representation for\(\displaystyle \frac{\sin(\pi z)}{\pi z}\) on p. 69. This is of course good news with a vengeance. By the way, a little more bad news: the Cauchy-Riemann equations occur on p. 33, which, again, is far too late in the game for my taste.

Now, to be fair, the book has a second part: on p. 89 the authors give us “Applicable Real and Complex Functions” and introduce what they’re about to do with the comment that “[t]his chapter serves [to supply a solid basis] for real and complex analysis abridging [sic] a gap between them, incorporating many applications of real analysis.” Here the authors go about developing in some detail what they sketched or hinted at earlier and, most importantly, provide much-needed proofs: Cauchy’s Integral Theorem on p. 125 (as already mentioned) and his Integral Formula on p. 130 — all’s well with the world at last. Additionally there are a lot of other things available in this part of the book that are certainly worth it. For example, the authors cover a weak version of the Phragmén-Lindelöf Theorem, due to E. C. Titchmarsh, on p. 131.

So there it is, then. This is a very interesting text with a great deal to recommend it. Since it would be psychologically impossible for me to present Cauchy’s Integral Theorem as a black box, I’d certainly start off with a derivation of the Cauchy-Riemann equations and then recall Green’s Theorem to get a proof early on (in fact, I generally go this route), and there other places where I’d either change the sequence a bit or add more structure. However, I think this book is very easily adaptable to the kind of course in complex analysis I described above, and which is all but ubiquitous these days. So, indeed, Chakraborty, Kanemitsu, and Kuzumaki have crafted a very useful text.

Two final quibbles: as could already be gleaned from the corrections needed to the quotes above, language is a problem, but not really a disqualifying one. And I’m afraid I am not a fan of the pencil-drawn portraits included in the book: if you’re going to go that route, use the wonderful old pictures that are available (and here is my all-time favorite).

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