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A Course of Modern Analysis

E. T. Whittaker and G. N. Watson
Cambridge University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on
See our review of the fourth edition. This is a re-typeset version of that edition. There is some additional material, but the body of the work is unchanged except for formatting. The new material is: (1) a short Foreword by S. J. Patterson putting the book in historical context; (2) a lengthy Introduction by Victor H. Moll, surveying the changes that have occurred in mathematical analysis since the last (1927) edition, and giving a detailed summary of each chapter; (3) all the references have been moved to the back and written out in full, in the modern style (the old style was to cite in footnote by author’s last name, journal name, and volume).
This new edition is much easier to read than the old one, and if you use this book at all that’s a good enough reason to buy a copy of the new one. One drawback of the new edition is that the re-keying introduced a large number of typographical errors (I marked about 100 errors in my copy, without trying very hard). Happily, most of these are obvious and so are unlikely to cause confusion; the worst one is replacing o(1) with O(1) in the conclusion of the Riemann-Lebesgue lemma (p. 177), causing us to assert that the Fourier coefficients are bounded rather than going to zero.
If you have not been using this book (or at least hearing about it), it would be hard to recommend that you start, even with this new edition.  The approach to analysis (the first half of the book) is antiquated, and although still correct, is generally not how people do analysis these days. There is no point-set topology (except for the Bolzano-Weierstrass theorem), and the book is usually vague about the exact meanings of regions and curves.  The integration is the Riemann integral (or more precisely the Darboux upper and lower integrals) rather than Lebesgue’s. The book is very careful about convergence and interchanging the order of sums and/or integrals, but the general approach is through complex analysis rather than measure and integral: happily, all the special functions of interest are analytic, so this works well.
The real value of the book is the second half, which is all about special functions. In many cases the coverage here is still the best or one of the best available, and is concise and all in one volume. I think this is true in particular for the gamma function, the theta functions, and the elliptic functions.

Allen Stenger is a math hobbyist and retired software developer. He was Number Theory Editor of the Missouri Journal of Mathematical Sciences from 2010 through 2021. His personal website is His mathematical interests are number theory and classical analysis.