This is the second volume of a series on analysis. The authors “assume as known the content of” the first volume (page v) and make occasional references to it, but they add that “whenever possible, we provide an alternative elementary treatment in order to allow the use of part of this volume on sequences and series, independently from infinitesimal calculus.” I did not notice any forward references to later volumes. I have previously reviewed volume 1 here, and a later volume has been reviewed here.

The authors are sometimes sloppy, and there are minor problems with language (they originally wrote in Italian), but these are good books, with lots of worthwhile mathematics over a broad range of topics. The main question I have in reading the first two volumes is one of audience: must they be relegated to the dustbin of supplementary reading, or can one imagine a course in which one or both could be used?

I have a very hard time doing this for the first volume alone, since it is essentially a calculus book at the advanced undergraduate level. It could work in an Advanced Calculus course for very strong sophomores who will later take a more ambitious analysis course, but how many institutions can offer that? And if yours could, why not use Hardy’s A Course of Pure Mathematics instead? But the first two volumes taken together contain nearly everything that would be in a semester course in undergraduate analysis, and a lot more. For a school that has good students but is short on upper level classes, they could be the basis for a sequence for math majors; I would hesitate to call it analysis 1 and 2, but students would get the equivalent of a semester or more of analysis and a smattering of other good things.

Volume 2 is a real hodgepodge that could only be the primary textbook for a course specifically based on it. I’m not sure what name it would have, but it would be an interesting course. Chapter 1 is on the real numbers, the natural numbers, and induction. It is not unlike the first chapter of a standard undergraduate analysis book such as Stephen Abbott’s Understanding Analysis or Jeffery Cooper’s Working Analysis, but with more emphasis on induction. It has Cauchy in the wrong century and a few other misprints, but on the whole I think it is better than Cooper’s first chapter. (I don’t mean to pick on him specifically, but I happen to be familiar with his book and I think the first half of it is pretty close to the industry average.)

As with volume 1, almost every chapter concludes with a page or two of “Summing Up”, followed by a large exercise set, and there are other exercises interspersed throughout the text. The frontispieces of many classic mathematical works are also here. An amusing example in chapter 1 is an Italian translation of Stetigkeit und irrationale Zahlen, the author of which, as we all know, is Riccardo Dedekind.

Chapter 2 is on sequences, again parallel to Abbott and Cooper, and again the book under review has some advantages. A standard (but nice) proof of Wallis’s formula, starting from the integral of the nth power of sin(x) from 0 to π/2, is on page 55. One would never see this in Abbott’s or Cooper’s chapter 2 because they have not discussed integration yet; Giaquinta and Modica have no such qualms because they did integration in volume 1. Wallis’s product is not only beautiful but also useful in completing a proof of Stirling’s formula (also not in Abbott or Cooper) on page 56. On the other hand, the argument of Example 2.60 on page 53 looks like something that a particularly dim student might think up, although it is easy to fix.

Section 3.3 has set theory and some of Cantor’s ideas, but in other respects chapter 3 looks like it wandered in from a discrete mathematics book. Section 3.1 is on elementary number theory, starting with the Euclidean algorithm and moving through congruences and Fermat’s little theorem to RSA encryption. Section 3.2 is on elementary combinatorics, particularly applications of binomial coefficients. Problem 3.96 has 10 simple binomial coefficient identities, 5 in each of two columns. The last one in the right column is wrong, and if it were true it would repeat the one above it. Not only is the third one in the left column also wrong, but I can’t even figure out what the authors were trying to write down.

Chapter 4, on the geometry of complex numbers, is short and could pass for the first chapter of an undergraduate complex analysis book. Chapter 5 is entitled “Polynomials, Rational Functions and Trigonometric Polynomials.” The first two sections are on the theory of equations, a nice old-fashioned subject that rarely appears in a modern book except for a small role in abstract algebra. The centerpiece of section 5.3 is Hermite’s decomposition of a rational function, which allows one to integrate even if one can’t find the factoring of the denominator explicitly. (Ostrogradsky’s name is often associated with this idea also.) The rest of the chapter is on finite Fourier series. It includes a proof that the integral of (sin t)/t over all positive t is π/2, which is similar to but not as good as the proof in section 1.76 of Titchmarsh’s classic The Theory of Functions.

In chapters 6 and 7 we come back to a more typical concern of analysis, infinite series. Chapter 6 is mostly standard, but there is a nice argument beginning on page 212 that leads to Euler’s infinite product for sin(x) and his partial fractions expansion of cot(x). A sign is wrong in the statement of the latter, and one of the early steps in the proof is misstated — one divides by sin(t), not by C, before letting t approach zero. In Example 6.5 one should multiply by (1–q)^{2}, not 1–q^{2}, and the result can be obtained more easily by differentiation. There is an obvious misprint in the statement of the comparison test on page 204. Other obvious misprints occur in Examples 6.14 and 6.40.

One of my pet peeves occurs near the beginning of chapter 7, where the authors define the Euler number E_{n} as the coefficient of x^{n}/n! in the expansion of sech(x). Many other authors do this, but it is a terrible definition both mathematically and historically. It makes half of the numbers zero, since sech(x) is an even function, while the other half are integers that alternate in sign. What one should do is define E_{n} as the coefficient of x^{n}/n! in the expansion of sec(x) + tan(x); since sec(x) is an even function and tan(x) an odd one, this decouples into expansions of each. The historical reason why this is often not done is that, as Euler already realized, tan(x) has a reasonably nice expansion in terms of Bernoulli numbers (so too do cot(x) and csc(x)), but sec(x) does not. Still, one gets a nicer expansion of tan(x) by treating it in parallel with sec(x). This is implicit in some of Euler’s work — see volume 14 of his *Opera Omnia*, 1st series, pages 416–417, where the first 14 Euler numbers can be found — but Scherk seems to have been the first to make it explicit, in his *Mathematische Abhandlungen* of 1825. (It is interesting to watch Scherk slowly groping his way toward this idea, perhaps inhibited by his admiration for Euler.) Now all the Euler numbers are positive integers, and they have a beautiful combinatorial interpretation which was found in 1879 by André: E_{n} counts the number of “up-down” permutations of {1,2,...,n}, i.e., those that alternately rise and fall, starting with a rise, so that, for example, the up-down permutations of {1,2,3,4} are 1324, 1423, 2314, 2413 and 3412, and hence E_{4} = 5. After some material on operations with power series the authors return to combinatorics in subsection 7.3.4, with a brief discussion of generating functions, but without this example.

Section 7.4 has the Euler-Maclaurin summation formula and the gamma and beta functions. The well-known application of Bernoulli polynomials to the sum of the first n m-th powers in the middle of page 277 is given with up to four errors: (i) the left side should stop at k = n and not go all the way to infinity; (ii) on the right side, n should be n + 1 (unless it was intended that the left side stop at n – 1); (iii) the last m on the right side should be m + 1; (iv) either the sum should start at k = 1, or the B_{m}(1) on the right side should be B_{m+1}(0) rather than B_{m+1}(1) — this matters if and only if m = 0.

Proposition 7.90 is fine, but the sentence that precedes it is misleading. Euler’s original definition of the gamma function is not the infinite product in the statement of the proposition, which is Weierstrass’s definition, but the infinite product at the end of the proof.

Problem 7.101 asks for the sums of 10 power series, another difference between this book and those of Abbott and Cooper. The last one, as stated, makes no sense because the summation index is misprinted, but one can see from a hint what the authors had in mind; the answer is log(sin x) – log(x). There are also several misprints in problem 7.115.

Even by this book’s standards the last chapter is eclectic. The first section starts with difference equations and proceeds through Newton’s Method and continued fractions to rational approximation. The last two sections are on discrete dynamical systems, concluding with chaos and fractals. The Cantor set appears here rather than in chapter 3. In the introduction the authors say that this final chapter “should be regarded as an invitation to further study.”

In this series Giaquinta and Modica have set themselves the formidable task of constructing from scratch an analysis sequence of several years length. Although unafraid to bring in modern ideas, they have more regard for classical topics and arguments than most authors writing analysis books today. If this suits your taste, as it does mine, then these books are well worth looking at. I enjoyed reading this volume a little more than volume 1, and I look forward to reading future volumes, if not to reviewing them.

Warren Johnson is visiting assistant professor of mathematics at Connecticut College.