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Mathematical Methods for Physicists

George B. Arfken, Hans J. Weber, and Frank E. Harris
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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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.

  1. Mathematical Preliminaries
  2. Determinants and Matrices
  3. Vector Analysis
  4. Tensors and Differential Forms
  5. Vector Spaces
  6. Eigenvalue Problems
  7. Ordinary Differential Equations
  8. Partial Differential Equations
  9. Green's Functions
  10. Complex Variable Theory
  11. Further Topics in Analysis
  12. Gamma Function
  13. Bessel Functions
  14. Legendre Functions
  15. Angular Momentum
  16. Group Theory
  17. More Special Functions
  18. Fourier Series
  19. Integral Transforms
  20. Periodic Systems
  21. Integral Equations
  22. Mathieu Functions
  23. Calculus of Variations
  24. Probability and Statistics