This is the first of several volumes on analysis by these authors. They were originally published in Italian, and then translated by the authors into English. The second volume is on Approximations and Discrete Processes, and a later volume on functions of several variables was also reviewed here. Anyone who studied these volumes carefully would certainly learn a lot of worthwhile mathematics, although not all of it would be analysis, as we shall see.

At the risk of sounding provincial, I will largely be concerned here with the utility of the first volume in courses at an American college or university. The authors have very little to say about who these volumes are intended for or what background the reader should have. The back cover of volume 1 calls it “an excellent resource for self-study or for classroom use at the advanced undergraduate or graduate level”, whereas volume 2’s back cover says it “may be used in graduate seminars and courses or as a reference text by mathematicians, physicists, and engineers.”

The book under review “starts with an introductory chapter at elementary level to be used according to the various degrees of competence acquired by students at school” (page vi; I don’t think “competence” is the best choice here, but the authors are after all writing in their second language). This chapter begins with some things we expect to see in a first chapter of a book entitled *Mathematical Analysis*, such as definitions of supremum and infimum, the completeness axiom, the triangle inequality, and a proof that √2 is irrational. Then it veers off into vectors in the plane and, briefly, “the space”, i.e. in three dimensions, and continues with lines and circles and a focus/eccentricity treatment of conic sections, before returning to familiar ground with a quick discussion of surjective and injective functions. Six pages on trigonometry follow, with only four trig functions, which is however one and a half more than Hughes Hallett et al. The authors know (page 41) that cot x = 1/tan x and that cot x = tan(π/2 – x), and that tan x tan(π/2 – x) = 1, yet somehow they can still say that cot x cot(π/2 – x) = –1. After a few words on functional composition and inverse functions we get the inverse trig functions (naturally only four of them), followed by exponentials and logarithms and a brief subsection on logarithmic coordinates.

At the end of each chapter is a large exercise set, with other exercises interspersed throughout the text. Each chapter except the first also has a concluding section called “Summing Up”, which takes less than a page for chapter 2 but runs to five and a half pages for chapter 5. The first chapter concludes instead with an interesting section entitled “Remarks on Common Language and the Language of Mathematics.” For example (page 50), “The article *the* identifies a person or an object univocally” (a legitimate English word, although a native speaker would probably use either uniquely or unequivocally), in contrast to “a” or “an”. A discussion of propositional calculus begins on page 52. We learn that “Mr. Smith is taller than Mr. Fire” is not a proposition because, not knowing which Mr. Smith or Mr. Fire it refers to, we cannot say whether it is true or false. (This reminded me of Nabokov’s novel *Pnin*.) The truth tables on page 54 have not been translated into English — we find V in place of T. There is a brief mention of the axiom of choice on page 57, where “self-convincing” is used in place of “self-evident”.

The problem with this book may already be apparent: who is the audience? It is essentially a calculus book at the advanced undergraduate level, or an analysis textbook for someone who has never seen calculus before. It has more than a few nice features, and I can just barely imagine a group of students that it would suit, but I don’t think I’ll ever meet them.

Chapter 2 is on limits and continuity. The epsilon-delta definition of limit is on page 66, and of continuity on page 78, and we find the intermediate value theorem on page 80. The other main result of the chapter is what the authors call Weierstrass’s theorem, that a continuous function on a closed and bounded interval attains its maximum and minimum. The authors are interested in history in the sense that they like to associate a name to a result (e.g., the law of cosines is Carnot’s formula), but they do not give references. The word “compact” is not mentioned, here or later.

Chapter 3 is entitled “The Fundamental Ideas of the Differential and Integral Calculus”. The ratio [f(x) – f(a)]/[x – a] is called a “differential quotient” rather than a difference quotient, and there is no attempt to reconcile this term with the definition of differential. The authors define the elasticity of f(x) as E(x) = xf'(x)/f(x), but they can’t give examples yet because they don’t compute any derivatives until chapter 4. They claim that E(x) is the derivative of log[f(e^{x})], which is true if e^{x} is replaced by x after the differentiation. This notion of elasticity is useful in economics.

Subsection 3.1.3 is entitled “Fermat and Lagrange theorems.” I invite the reader to guess what Lagrange’s theorem is. Fermat’s theorem is probably familiar: if f is differentiable at a local maximum or minimum point m that is not an endpoint, then f'(m) = 0. The authors make a very interesting remark generalizing this: if m is a relative minimum for f on an interval I, then f'(m)(m – x) is nonpositive for all x near m and in I. (This covers endpoints and interior points.) Lagrange’s theorem is the mean value theorem, and the latter name is usually used after this subsection, though not always. Results like the mean value theorem can be found in several places in Lagrange’s two sets of lectures on calculus, *Théorie des Fonctions Analytiques* and *Leçons sur le Calcul des Fonctions*, which are volumes 9 and 10 respectively in his Oeuvres. See for example page 239 of the former or page 87 of the latter. Lagrange’s form of Taylor’s theorem with remainder is, of course, a generalized mean value theorem.

After some standard applications of the mean value theorem, chapter 3 proceeds to the Riemann integral, the fundamental theorem of calculus, and the mean value theorem for integrals. It concludes with a section of about 7 pages of historical remarks. A notable feature of the book is the reproduction of the frontispieces of many classic (and some not so classic) mathematical works, and these are particularly dense in this section. They’re kind of fun to glance at. There is a rather strange remark near the bottom of page 136: “the notion of limit developed only in the nineteenth century”. I would rather say that calculus was not systematically based on the notion of limit until the nineteenth century. D’Alembert was already advocating this in the middle of the eighteenth century.

The mechanics of differentiation and integration are dealt with in chapter 4. In the brief discussion of hyperbolic functions on page 151 are two instances of a misprint that is common among students seeing them for the first time, but surprising in this context: writing sin hx for sinh x. On page 168 the authors mention the substitution t = tan(x/2) for rational functions of sine and cosine, but note “that it is more convenient to use one of the changes of variable t = sin x, t = cos x or t = tan x provided it leads to the integration of a rational function,” for example the integral of sin x/(sin x + cos x), where presumably we are to use t = tan x. Better is to observe that there are two integrals of this type that are easy, namely (cos x + sin x)/(cos x + sin x) and (cos x – sin x)/(cos x + sin x), and any integral of the form (Acos x + Bsin x)/(cos x + sin x) is a linear combination of these.

The authors stress the distinction between indefinite integrals and primitives. The substitution t = tan(x/2) is reasonably effective at integrating 1/(2+sin x), giving the answer (2/√3) times the arctangent of (2 tan(x/2) + 1)/(√3). But this is not a primitive of 1/(2+sin x) on (0,2π) because it is not continuous at π, an excellent point that few authors bother to make explicit. What then should we do if we have to integrate 1/(2+sin x) from 0 to 2π? Denoting the above antiderivative by f(x), the authors’ answer is that we should define a new function g(x) to agree with f(x) from 0 to π, to equal π/(√3) at π, and to equal f(x) + (2π)/(√3) from π to 2π. Then g(x) is continuous from 0 to 2π, so we can just plug in the endpoints. They might have pointed out in addition that by periodicity, the integral from 0 to 2π is the same as that from –π to π, and on the latter interval f(x) is a perfectly good primitive. Having observed this, we can also take twice the integral from –π/2 to π/2, and if I personally had to integrate 1/(2+sin x) from 0 to 2π, I would first make this reduction and then substitute sin u = (1+2 sin x)/(2+sin x). Try it, it’s fun!

Next comes a section where the arctangent and the natural logarithm (and hence the other elementary transcendental functions) are redefined by definite integrals. I won’t try to reproduce their definition of arctangent, formula (4.13) on page 170, but it isn’t the one you would probably expect. An alternate form that allegedly comes from changing variables in (4.13) is at the top of page 171, but it is wrong on two counts. It should be what you get by changing x to 1/x in (4.13) and then subtracting it from π/2. Once the arctangent is defined, the tangent is its inverse, and then sine and cosine are defined by using their expressions in terms of tan(x/2). The derivatives of sine and cosine are now to be computed from these expressions! With these in hand, the authors give a nice calculus proof of the addition formula for cosine on page 172. They give a similar proof of a special case first, sin x=cos(π/2 + x). Unfortunately, sin x doesn’t equal cos(π/2 + x); rather cos x=sin(π/2 + x).

Section 4.4 is again reminiscent of a calculus book, with a discussion of some simple differential equations. I am puzzled by the comment following (4.26) on page 180 (the general solution of the logistic equation) that the particular solutions y = 0 and y = L are not included in the general solution, as they are the cases c = 0 and c=∞ respectively. Now section 4.5 really looks like analysis: a painstaking effort to extend the Riemann integral to improper integrals, by distinguishing measurable, summable, and integrable functions. Chapter 5 has a similar flavor. It starts with Taylor’s theorem and moves on to convexity and inequalities (Bernoulli, Cauchy, Young, Holder, Minkowski, Jensen; also Clarkson, which was new to me) before finishing with some remarks on graphing.

The translation from Italian is pretty good most of the time but is not impeccable, as we have seen, and on page 230 is another example: “In this section we collect a few inequalities of general use. They can be often proved by other means, but usually the methods of calculus, and especially convexity, yield to simpler and more direct proofs.” The word “to” is contrary to what the authors are trying to say — they think convexity is the best tool here and they mean to use it whenever they can.

If you are ever in need of a nasty limit problem, there are scads of them in the exercises on pages 257–258; I spent a pleasurable couple of hours working many of them out. Two are repeated, and a few others could be improved. For example, problem 5.122 asks for constants a and b such that two particular limits should work out to 1/2. In the first of these 1/3 would be a better right side. In that case a = –1 and b = 1, whereas you have to solve a^{3} + a = –3 if the right side is 1/2.

The sixth and final chapter is the most eclectic. It begins with two sections on differential equations, at a somewhat higher level than section 4.4, with applications to classical mechanics in a third section. The final section is on optimization, but with a vastly different emphasis than in an American calculus course. A subsection on isoperimetric problems leads ultimately to several pages on graph theory. Only the material on differential equations appears in the chapter summary.

I don’t want to give a wrong impression — I like this book, and I would like it even more if it were cleaned up a little. I can’t recommend it as a textbook, but there are some things I may want to refer to it for the next time I teach analysis.

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