You are here

Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions

George Pólya and Gabor Szegö
Springer Verlag
Publication Date: 
Number of Pages: 
Classics in Mathematics
Problem Book
BLL Rating: 

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

[Reviewed by
Allen Stenger
, on

These are famous but old problem books, originally published in German in 1925, then lightly revised several times and then published in slightly expanded English editions in 1972 and 1976. By “analysis” they mean what was in the mainstream of analysis in 1925, that is, mostly theory of functions of a single complex variable. They do include many related matters, such as sequences, Riemann integrals, asymptotics, and more, but they omit more modern analysis topics such as functional analysis, linear spaces, and measure theory. They also give more than a little coverage of topics that were hot back then but have dimmed some today, such as equidistribution of sequences and of schlicht (univalent) functions.

As in Pólya’s other books, the main concern is discovery and problem solving rather than mathematical facts. An important distinction from most problem books is that few problems appear in isolation, but are nearly always within a sequence of problems that builds on one idea and explores its consequences. The Preface goes into quite a lot of detail about how to use the book, and it includes much useful advice about learning mathematics in general. (It is the source of the famous saying “An idea which can be used only once is a trick. If one can use it more than once it becomes a method.”)

Each problem has a solution given, although very briefly. A little over half of the page count is devoted to solutions, so on the average the solution is only a little longer than the problem statement itself. In a few cases the solution is not given and there is instead a reference where it can be found in the literature.

Despite the age of the material, I think these books are still very valuable as references, and I have used them several times when solving problems in the American Mathematical Monthly. The indices are skimpy, so often the best way to find something is to go to the relevant section and just skim through all the problems. On the plus side, the solutions are generally well-documented with references to the literature, and the books are also extensively cross-referenced when there is a related problem or technique in another section. Most of the literature references are carried over from the original books, so they are to works published before 1925, although there are scattered references to newer work also.

Some examples of things covered here that are hard to find elsewhere are representing and evaluating limits as a Riemann integral, the Lagrange reversion for power series, and asymptotics by the saddle-point method. There is also an extremely thorough coverage of Descartes’s Rule of Signs, including not only several looks at why it works but some generalizations as well.

In my opinion the problems are very difficult by today’s standards, and apparently this was true in 1925 as well. J. D. Tamarkin’s review in Bulletin of the AMS in 1928 said “It is the authors’ [i.e., Pólya and Szegő] teaching experience that each chapter can be worked through in an advanced class in one semester (2 hours per week); this certainly requires that the students be excellently prepared.” There are 29 chapters, so we are talking about covering around 15 pages of problems per semester.

Bottom line: Still relevant and valuable after all these years.

See also the page for volume II.

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

Infinite Series and Infinite Sequences

1 Operations with Power Series
Additive Number Theory, Combinatorial Problems, and Applications
Binomial Coefficients and Related Problems
Differentiation of Power Series
Functional Equations and Power Series
Gaussian Binomial Coefficients
Majorant Series

2 Linear Transformations of Series. A Theorem of Cesàro
Triangular Transformations of Sequences into Sequences
More General Transformations of Sequences into Sequences
Transformations of Sequences into Functions. Theorem of Cesàro

3 The Structure of Real Sequences and Series
The Structure of Infinite Sequences
Convergence Exponent
The Maximum Term of a Power Series
Rearrangement of the Terms
Distribution of the Signs of the Terms

4 Miscellaneous Problems
Enveloping Series
Various Propositions on Real Series and Sequences
Partitions of Sets, Cycles in Permutations


1 The Integral as the Limit of a Sum of Rectangles
The Lower and the Upper Sum
The Degree of Approximation
Improper Integrals Between Finite Limits
Improper Integrals Between Infinite Limits
Applications to Number Theory
Mean Values and Limits of Products
Multiple Integrals

2 Inequalities
Some Applications of Inequalities

3 Some Properties of Real Functions
Proper Integrals
Improper Integrals
Continuous, Differentiate, Convex Functions
Singular Integrals. Weierstrass’ Approximation Theorem

4 Various Types of Equidistribution
Counting Function. Regular Sequences
Criteria of Equidistribution
Multiples of an Irrational Number
Distribution of the Digits in a Table of Logarithms and Related Questions
Other Types of Equidistribution

5 Functions of Large Numbers
Laplace’s Method
Modifications of the Method
Asymptotic Evaluation of Some Maxima
Minimax and Maximin

Functions of One Complex Variable. General Part

1 Complex Numbers and Number Sequences
Regions and Curves. Working with Complex Variables
Location of the Roots of Algebraic Equations
Zeros of Polynomials, Continued. A Theorem of Gauss
Sequences of Complex Numbers
Sequences of Complex Numbers, Continued: Transformation of Sequences
Rearrangement of Infinite Series

2 Mappings and Vector Fields
The Cauchy-Riemann Differential Equations
Some Particular Elementary Mappings
Vector Fields

3 Some Geometrical Aspects of Complex Variables
Mappings of the Circle. Curvature and Support Function
Mean Values Along a Circle
Mappings of the Disk. Area
The Modular Graph. The Maximum Principle

4 Cauchy’s Theorem • The Argument Principle
Cauchy’s Formula
Poisson’s and Jensen’s Formulas
The Argument Principle
Rouche’s Theorem

5 Sequences of Analytic Functions
Lagrange’s Series. Applications
The Real Part of a Power Series
Poles on the Circle of Convergence
Identically Vanishing Power Series
Propagation of Convergence
Convergence in Separated Regions
The Order of Growth of Certain Sequences of Polynomials

6 The Maximum Principle
The Maximum Principle of Analytic Functions
Schwarz’s Lemma
Hadamard’s Three Circle Theorem
Harmonic Functions
The Phragmén-Lindelöf Method