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Publisher:

Dover Publications

Publication Date:

2007

Number of Pages:

607

Format:

Paperback

Price:

29.95

ISBN:

9780486462516

Category:

Textbook

[Reviewed by , on ]

Allen Stenger

04/26/2008

This is a rigorous single-variable calculus book with a myriad of challenging, non-routine exercises. It is an unaltered reprint of the 1963 Pergamon edition.

This is a pure mathematics version of calculus, with no applications and hardly any analytic geometry. The transcendental functions are defined by power series and not geometrically or as integrals. Rankin's treatment is notable for a lengthy development of the much-neglected but very useful Riemann-Stieltjes integral. He introduces the limit process by working first on sequences (instead of functions of a real variable), an approach I like very much. There's quite a lot of formulaic manipulation of integrals and of infinite series, and several advanced convergence tests.

Two similar books are:

- Hardy's A Course of Pure Mathematics is very similar in tone and coverage. Rankin is more encyclopedic and has many more exercises. Hardy has much clearer explanations. Surprisingly, Hardy does not deal with the Riemann integral at all! An integral to him is an anti-derivative, and can also be visualized as the area under the curve, but is not a limit of anything. Hardy's book was credited with revolutionizing the practice of analysis in England when in came out in 1908. It's almost incredible to look at it today and realize that it's really just a calculus book, and one that doesn't even have an integration theory.

- Landau's Differential and Integral Calculus is another rigorous calculus book. The mathematical approach is similar to Rankin, but this is a much more concise book and covers many fewer topics. There are no exercises. Landau's books in general, including this one, are noted for their very slick and clear proofs.

None of these three books is what we today would call a real analysis book: they have no topology, they don't treat functions as objects, there's no possibility of extending anything to more general spaces, and they take the real number system for granted.

Which raises the question: Is there still a market for this kind of book? Probably no one teaches a real analysis course today without a lot of point-set topology and metric spaces. Calculus courses in the US have veered away from being "proof" courses, partly from a desire to emphasize applications and partly because students are not on the average as well prepared as they used to be and so tend to be baffled by proofs.

On the other hand, Dover apparently thinks there's still some kind of market out there, since they've just selected the book for reprinting. Hardy's book has been in print continuously for 100 years and has just been issued in a special Centenary edition. Somebody must be buying it, although I suspect they are not students but professional mathematicians who admire and appreciate Hardy's treatment.

I admire Hardy's and Landau's treatments. I'm not as enthusiastic about Rankin's: the exposition is a bit choppy and at times hard to follow, and the variety of topics can be overwhelming. He does have great exercises, and if you are teaching real analysis, or a calculus course for bright students, it's worth having a copy of Rankin at hand as a source of challenges for your students.

Allen Stenger is a math hobbyist, library propagandist, and retired computer programmer. He volunteers in his spare time at MathNerds.com, a math help site that fosters inquiry learning. His mathematical interests are number theory and classical analysis.

Preface

List of Symbols and Notations

1. FUNDAMENTAL IDEAS AND ASSUMPTIONS

1. Introduction

2. Assumptions relating to the field operations

3. Assumptions relating to the ordering of the real numbers

4. Mathematical induction

5. Upper and lower bounds of sets of real numbers

6. Functions

II. LIMITS AND CONTINUITY

7. Limits of real functions on the positive integers

8. Limits of real functions of a real variable x as x tends to infinity

9. Elementary topological ideas

10. Limits of real functions at finite points

11. Continuity

12. Inverse functions and fractional indices

III. DIFFERENTIABILITY

13. Derivatives

14. General theorems concerning real functions

15. Maxima, minima and convexity

16. Complex numbers and functions

IV. INFINITE SERIES

17. Elementary properties of infinite series

18. Series with non-negative terms

19. Absolute and conditional convergence

20. The decimal notation for real numbers

V. FUNCTIONS DEFINED BY POWER SERIES

21. General theory of power series

22. Real power series

23. The exponential and logarithmic functions

24. The trigonometric functions

25. The hyperbolic functions

26. Complex indices

VI. INTEGRATION

27. The indefinite integral

28. Interval functions and functions of bounded variation

29. The Riemann-Stieltjes integral

30. The Riemann integral

31. Curves

32. Area

VII. CONVERGENCE AND UNIFORMITY

33. Upper and lower limits and their applications

34. Further convergence tests for infinite series

35. Uniform convergence

36. Improper integrals

37. Double series

38. Infinite products

HINTS FOR SOLUTIONS OF EXERCISES

INDEX

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