Alf van der Poorten accurately summarized this book in his Notes on Fermat's Last Theorem by writing, “Notwithstanding its title, ‘Whittaker and Watson’ is neither ‘modern’ nor ‘analysis,’ as we now understand it. But it is the bible of the classical special functions.”
The last edition of this bible was in 1927, and it is still in print from Cambridge. Although it covers a lot of ground, its focus is primarily on the differential equations of mathematical physics and the special functions that are their solutions. This is not a mathematical physics book — it quotes the differential equations and states where they arise, but does not derive them or show how to solve them. The book is notable for having a large number of very challenging exercises (here called “Examples,” in the traditional English style).
The book is divided in two parts. Part I, “The Processes of Analysis,” starts at the beginning and presents all the background needed for Part II, “The Transcendental Functions.” Part I covers a large swath of classical analysis in only 250 pages, including the Riemann integral, infinite series, analytic functions, quite a lot about Fourier series, and some about differential and integral equations. It is not comprehensive on any of these subjects, but it covers the basics thoroughly and some of the more advanced parts. The treatment is straightforward and not too different from what is in most contemporary introductory texts on these subjects.
Part II, at about 350 pages, gives an almost-comprehensive account of the special functions of mathematical physics, and of a number of other special functions that are used more generally, for example, the gamma function and the elliptic functions. There is a relatively skimpy chapter on the Riemann zeta function. Most or all of these functions have individually been the subjects of whole books, but Whittaker and Watson give you coverage that is almost as complete, and often easier to use, in only 30 or 40 pages. Part II, used as a reference rather than as a text, is the real strength of this book.
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.
Part I. The Processes of Analysis
1. Complex numbers
2. The theory of convergence
3. Continuous functions and uniform convergence
4. The theory of Riemann integration
5. The fundamental properties of analytic functions
6. The theory of residues
7. The expansion of functions in infinite series
8. Asymptotic expansions and summable series
9. Fourier series and trigonometrical series
10. Linear differential equations
11. Integral equations
Part II. The Transcendental Functions
12. The gamma function
13. The zeta function of Riemann
14. The hypergeometric function
15. The Legendre function
16. The confluent hypergeometric functions
17. Bessel functions
18. The equations of mathematical physics
19. Mathieu functions
20. Elliptic functions
21. The theta functions
22. Jacobian elliptic functions
23. Ellipsoidal harmonics and Lamé's equation
Appendix: The Elementary Transcendental Functions
List of Authors Quoted