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A Treatise on Advanced Calculus

Philip Franklin
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is an introductory course in real analysis that provides the theoretical background for courses in single- and multi-variable calculus and series, but does not explore much beyond that. It is not a “rigorous calculus” course, because it assumes you have already had calculus and just need to understand the proofs behind calculus. It is strictly a proofs book and does not give examples (you are supposed to recall these from your calculus course), except for an occasional counterexample. The present volume is a Dover 2016 corrected reprint of the 1940 Wiley edition. It has held up very well over the years and neither the language nor the choice of results nor the proofs would seem strange in a new book today.

It is a very thorough course and covers more topics than would be in most calculus sequences. Some topics that would be new to most second-year students include the rudiments of complex variables and analytic functions, the theory of Fourier series, some fairly advanced existence theorems for differential equations, and a chapter on the gamma function. There’s just a little bit about elliptic functions, in the context of arc length. One unusual feature is that the trigonometric, exponential, and logarithmic functions are developed from the functional equations they satisfy instead of from geometry or power series.

The exercises are reasonably difficult and thorough, although for the most part not very challenging. They cover additional theorems (usually well-known) in the subject that did not make it into the main exposition, and some ask about properties of specific series or integrals. The more difficult exercises include sketches of their proofs. A few quite difficult theorems are buried in the exercises; my favorite is G. H. Hardy’s 1910 Tauberian theorem that a Cesàro-summable series whose terms are\(O(1/n)\) is actually convergent (exercise 40 on p. 512).

Unlike most real analysis courses, this one does not explore much beyond what is in calculus, but just tries to explain why calculus is true. Thus there is (for example) only a tiny bit of point-set topology, and no Lebesgue integration. It does begin with a construction of the real numbers (based on Dedekind cuts). It’s close in spirit to Ken Ross’s Elementary Analysis: The Theory of Calculus, although Ross’s book only covers single-variable calculus, and although Ross’s book does sneak in a few advanced topics that would not have been seen in calculus. A comparable book, but not as similar, is Apostol’s Mathematical Analysis.

Bottom line: still a very useful book, although it not what most people today would call “advanced calculus” or “real analysis”. The author’s intent seems to have been to provide a bridge course between the typical cookbook calculus course and the typical cours d’analyse that started at the graduate level. We usually take different development paths today and the book might not fit anywhere in your curriculum.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.


I. Real Numbers

II. Limits of Functions

III. Exponential, Logarithmic, and Trigonometric Functions

IV. Differentiation

V. Complex Numbers

VI. Integration

VII. Integrable Functions

VIII. Extensions and Applications of Integration

IX. Infinite Series and Infinite Products

X. Partial Differentiation

XI. Multiple Integration

XII. Sequences of Functions

XIII. Functions of Complex Variables

XIV. Fourier Series and Integrals

XV. Differential Equations

XVI. The Gamma Function and Other Definite Integrals