An Introduction to Mathematical Analysis

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.