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Publisher:

Springer Verlag

Publication Date:

2004

Number of Pages:

389

Format:

Paperback

Series:

Classics in Mathematics

Price:

49.95

ISBN:

978-3-540-63640-3

Category:

Problem Book

The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by , on ]

Allen Stenger

11/26/2014

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.

**One
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

Subseries

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

**Two
Integration**

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

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

**Three
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

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