This book is a completely rigorous treatment of calculus. It might be called “Pure Calculus” because there are no applications and it treats calculus as a subject worthy of study in itself. The book was written in 1934, based on Landau’s courses at Göttingen, was translated into English in 1950, and is still in print in 2009.

The book starts out, not with epsilons and deltas, but with limits of sequences. It has a very interesting (although far from intuitive) development of the natural logarithm of x as the limit of k(x^{(1/k)} – 1) as k goes to infinity through the powers of 2. It’s surprisingly easy to prove all the familiar properties of logarithms (without any reference to a general exponential), and then define the number e and the general exponential in terms of logarithms.

Power series are developed early, and used to define the transcendental functions. There’s just a little bit of several-variable calculus: partial derivatives are introduced, but only in support of the implicit function theorem. Integration is handled relatively briefly, and apart from the integration of rational functions there’s not a lot about techniques of integration.

The book has some goodies rarely found in calculus books: the Weierstrass approximation theorem, a proof of the fundamental theorem of algebra, a proof that rational functions can always be expressed in terms of partial fractions, the basic properties of the gamma function (including Euler’s reflection formula), and some existence theorems for Fourier series.

The book includes a large number of counterexamples. Whenever the bigger theorems are stated and proven, Landau usually shows that all the hypotheses are needed by dropping each one and giving a counterexample. In some cases the counterexamples are very elaborate, such as van der Waerden’s continuous, nowhere differentiable function, and a continuous function whose Fourier series diverges.

The calculus market has changed a lot since 1934, and by present-day standards this is not even a textbook: it has no exercises, no applications, and not even any drawings! What it does have, though, is clear and concise proofs of everything in single-variable calculus. This book, and G. H. Hardy’s Pure Mathematics, are the ones I turn to when I need a clear and rigorous proof of a calculus fact.

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.