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Theory of Functions

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This book is in the MAA's basic library list.

MAA Review

This is a very clearly-written text covering primarily complex analysis, and including a number of hard-to-find results. This second edition was published in 1939 and has been in print continuously since then.

The mathematical prerequisites are low: a rigorous course in single-variable calculus is enough. The book refers throughout to particular sections of Hardy’s A Course of Pure Mathematics for background (Hardy’s book, despite its very general title, is merely a rigorous calculus book). The most valuable sections for modern readers are the numerous far-from-trivial examples (throughout the book), an in-depth exploration of the maximum modulus theorem (including Hadamard’s three-circle theorem and the Phragmén-Lindelöf theorem), a thorough study of entire functions (here called integral functions), and a detailed look at asymptotics and Tauberian theorems for power series.

This is very much a pure-mathematics book, with no applications given to other sciences and almost no applications to other areas of mathematics. It’s a peculiar feeling to read a detailed chapter on Dirichlet series and to realize that it never mentions their use in number theory! It is also a very analytical book, with no pictures or geometric arguments, not even in the chapter on conformal mappings: everything is done with formulas.

The three chapters on Lebesgue measure and integration are probably the least valuable part of the book today. This theory has been streamlined and now has better terminology, and the material here is found in most books on real analysis in a more concise and easier-to-understand form. The one chapter of Fourier analysis is less dated, but the material is found in many real analysis books, and in most Fourier analysis books that use the Lebesgue integral.

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.

BLL** — The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

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