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The Integration of Functions of a Single Variable

G. H. Hardy
Dover Publications
Publication Date: 
Number of Pages: 
Dover Phoenix Editions
[Reviewed by
Henry Ricardo
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G. H. Hardy (1877-1947) was one of the most respected and influential analysts/number theorists of the first half of the twentieth century. The volume under review is a hardcover reprint of his 1916 classic tract. I use Hardy’s own word “tract” because this is not a textbook: There are no exercises and no applications (of course!).

In six elegant and precisely written chapters (Charles Dickens rather than Dan Brown), Hardy leads the reader through various classes of functions (see the Table of Contents), focusing on finding antiderivatives — or, as Hardy puts it, solving the equation dy/dx = f(x). In analyzing such an equation, Hardy is interested in “the functional form of y when f(x) is a function of some stated form.” He does not discuss the theory of integration (Riemann sums and so forth). Chapter VI contains a nice discussion of Liouville’s results on integration in finite terms. The Bibliography, consisting of references in French, German, and English, is history preserved in amber.

This is a book to have in the library as a reference; but its usefulness on one’s personal bookshelf has been superseded by many research and expository journal articles and by computer-oriented books such as Handbook of Integration by D. Zwillinger and the recent Irresistible Integrals by Boros and Moll.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.


2. Elementary Functions and Their Classifications
3. The Integration of Elementary Functions. Summary of Results
4. The Integration of Rational Functions
5. The Integration of Algebraical Functions
6. The Integration of Transcendental Functions
Appendix I. Bibliography
Appendix II. On Abel's Proof of the Theorem of v., §11