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Introduction to Calculus and Classical Analysis

Omar Hijab
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Kenneth A. Ross
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This is a revision of the first edition, published in 1997. The table of contents is almost identical to that of the first edition, but several sections have been rewritten. There are many unusual features in this book, so people considering it for an undergraduate course will want to carefully examine the book before making their decision.

One unusual feature is the "systematic avoidance of ε-δ arguments." For sequences, limits are first defined for monotone sequences using inf and sup. Then limits are defined for arbitrary sequences using limsup and liminf, which the author calls upper limits and lower limits. A sequence (an) is defined to be a Cauchy sequence provided there exists a sequence (εn) of positive numbers such that

lim εn = 0 and |an+m - an| ≤ εn

for positive integers m and n. For functions f, limx→cf(x) = L if for every sequence (xn) converging to c (where all xn≠ c), we have limn→∞f(xn) = L.

Another feature is to define integrals in terms of area: "Since the integral is supposed to be the area under the graph, why not define it that way?" To accomplish this, a detailed 15-page section is devoted to area in the plane. In the beginning, only rectangles whose sides are parallel to the axes are regarded as rectangles. These special rectangles are the building blocks for areas, i.e., the outer measures for all subsets of the plane.

The integral of any nonnegative function f (defined from a to b) is defined to be the area of its "subgraph," and the function is said to be integrable if the value is finite. Casual readers of the book should be aware that the symbol ∫f(x)dx is used even if the function f isn't measurable or integrable in the sense of Lebesgue.

Let's look at the status in this book of the basic and relatively simple fact about Riemann integrable functions:

∫[f(x)+g(x)]dx = ∫f(x)dx + ∫g(x)dx.

Exercise 4.3.5 asks the reader to use "additivity" to show this for nonnegative f and g where g is piecewise constant! In this exercise, "additivity" refers to additivity of areas; additivity for integrals refers to additivity over intervals: the integral on [a,b] is equal to the sum of the integral over [a,c] and the integral over [c,b]. The property under discussion is called "linearity" (of integrals). The solution to Exercise 4.3.5, where g is constant, is strained. Later, Theorem 4.4.5 establishes "linearity" for continuous f and g that are both nonnegative or both integrable… using the Fundamental Theorem of Calculus!

Armed with the Fundamental Theorem of Calculus, it is shown that the area of the unit disk is π. The number π is defined so that π/2 is the smallest positive solution of sin(x) = 1 where sin(x) is defined by its power series. This reader is unable to find arc length discussed in the book, even for the circumference of the unit circle. However, Theorem 3.5.1 identifies the coordinates of points on the unit circle with cos and sin, from which it follows that π/2 is the length of one-quarter of the unit circle. Thus the circumference of the unit circle has length 2π, confirming Archimedes' connection between the area and circumference of a circle.

The author is pleased that with his treatment of integration, "uniform convergence and uniform continuity can be dispensed with. Nevertheless, we give a careful treatment of uniform continuity, and use it, in the exercises, to discuss an alternate definition of the integral that was important in the nineteenth century (the Riemann integral)." Uniform convergence doesn't seem to appear in the book at all, though several theorems allow for the interchange of infinite sums and integrals.

The last chapter is full of interesting applications, among them Euler's gamma function, Stirling's approximation of n!, infinite products, Jacobi's theta function, and Riemann's zeta function.

I find the unusual features in this book interesting, but I would be very reluctant to impose them on students in an undergraduate one-variable analysis course. I especially feel that doing measure-theoretic area, in order to develop integration, is inappropriate at this level. Also, there are too many personalized definitions. Finally, I object to books that provide answers to all of (or none of) the exercises; instructors should have a choice, with answers provided to some of the exercises.

Kenneth A. Ross ( taught at the University of Oregon from 1965 to 2000. He was President of the MAA during 1995-1996. Before that he served as AMS Associate Secretary, MAA Secretary, and MAA Associate Secretary. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).

 The Set of Real Numbers.- Continuity.- Differentiation.- Integration.- Applications.- Solutions.- References.- Index.