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:
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.
List of Symbols and Notations
1. FUNDAMENTAL IDEAS AND ASSUMPTIONS
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
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
12. Inverse functions and fractional indices
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
27. The indefinite integral
28. Interval functions and functions of bounded variation
29. The Riemann-Stieltjes integral
30. The Riemann integral
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