Mathematical Methods for Physicists

This is a thorough handbook about mathematics that is useful in physics. It is a venerable book that goes back to 1966; this seventh edition (2012) adds a new co-author, Frank E. Harris. It is pitched as a textbook but because of its size and breadth probably works better as a reference. Very Good Feature: all the examples are real problems from physics. Note that it is not a mathematical physics book, though: it quotes the models but does not develop them.

For the most part the content is severely classical and the mathematics involved hasn’t changed much over the past century. Some newer topics include a lot of material that is useful for quantum physics, a little bit about signal processing and the discrete Fourier transform, and a unified treatment of vector spaces that includes function spaces and Hilbert spaces.

There are numerous exercises (the Preface claims nearly 1400), but they are uneven. Many are straightforward drill (and the answer is often given), but many are to “show” something. This seems peculiar in what is essentially a cookbook that omits most proofs.

The level of coverage is uneven. It is very thorough on differential equations and on special functions (as it should be). The chapter on complex variables is almost an entire course in itself. At the other extreme, some topics are so skimpy that they are little more than definitions and examples; these include tensors, line and surface integrals, and other coordinate systems. And there’s one oddball chapter all about angular momentum (in the quantum physics sense); it is the only chapter that is not about a mathematical technique. It is a very nice chapter, and may have been included to showcase some earlier techniques (specifically differential equations and tensors), but it appears without warning and without motivation.

There is an online companion site, that offers three additional chapters that did not fit in the book, along with a compendium chapter on infinite series compiled from material that is in the book.

A competing book, that is aimed at engineers but would also be useful to physicists, is Kreyszig’s Advanced Engineering Mathematics. It has more depth and less breadth, but also draws its examples mostly from physics. Another well-regarded book, that I have not seen, is Mary L. Boas’s Mathematical Methods in the Physical Sciences. This is pitched lower and covers the same topics as the present book, but not as thoroughly.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.