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Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry

George Pólya and Gabor Szegö
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Classics in Mathematics
Problem Book
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The Basic Library List Committee considers this book essential for 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 I.

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.

Functions of One Complex Variable. Special Part

1. Maximum Term and Central Index, Maximum Modulus and Number of Zeros
§ 1 (1–40) Analogy between μ(r) and M(r), ν(r) and N(r)
§ 2 (41–47) Further Results on μ(r) and ν(r)
§ 3 (48–66) Connection between μ(r), ν(r), M(r) and N(r)
§ 4 (67–76) μ(r) and M(r) under Special Regularity Assumptions

2. Schlicht Mappings
§ 1 (77–83) Introductory Material
§ 2 (84–87) Uniqueness Theorems
§ 3 (88–96) Existence of the Mapping Function
§ 4 (97–120) The Inner and the Outer Radius. The Normed Mapping Function
§ 5 (121–135) Relations between the Mappings of Different Domains
§ 6 (136–163) The Koebe Distortion Theorem and Related Topics

3. Miscellaneous Problems
§ 1 (164–174.2) Various Propositions
§ 2 (175–179) A Method of E. Landau
§ 3 (180–187) Rectilinear Approach to an Essential Singularity
§ 4 (188–194) Asymptotic Values of Entire Functions
§ 5 (195–205) Further Applications of the Phragmén-Lindelöf Method
§ 6 (*206–*212) Supplementary Problems

The Location of Zeros

1. Rolle’s Theorem and Descartes’ Rule of Signs
§ 1 (1–21) Zeros of Functions, Changes of Sign of Sequences
§ 2 (22–27) Reversals of Sign of a Function
§ 3 (28–41) First Proof of Descartes’ Rule of Signs
§ 4 (42–52) Applications of Descartes’ Rule of Signs
§ 5 (53–76) Applications of Rolle’s Theorem
§ 6 (77–86) Laguerre’s Proof of Descartes’ Rule of Signs
§ 7 (87–91) What is the Basis of Descartes’ Rule of Signs?
§ 8 (92–100) Generalizations of Rolle’s Theorem

2. The Geometry of the Complex Plane and the Zeros of Polynomials
§ 1 (101–110) Center of Gravity of a System of Points with respect to a Point
§ 2 (111–127) Center of Gravity of a Polynomial with respect to a Point. A Theorem of Laguerre
§ 3 (128–156) Derivative of a Polynomial with respect to a Point. A Theorem of Grace

3. Miscellaneous Problems
§ 1 (157–182) Approximation of the Zeros of Transcendental Functions by the Zeros of Rational Functions
§ 2 (183–189.3) Precise Determination of the Number of Zeros by Descartes’ Rule of Signs
§ 3 (190–196.1) Additional Problems on the Zeros of Polynomials

Polynomials and Trigonometric Polynomials

§ 1 (1–7) Tchebychev Polynomials
§ 2 (8–15) General Problems on Trigonometric Polynomials
§ 3 (16–28) Some Special Trigonometric Polynomials
§ 4 (29–38) Some Problems on Fourier Series
§ 5 (39–43) Real Non-negative Trigonometric Polynomials
§ 6 (44–49) Real Non-negative Polynomials
§ 7 (50–61) Maximum-Minimum Problems on Trigonometric Polynomials
§ 8 (62–66) Maximum-Minimum Problems on Polynomials
§ 9 (67–76) The Lagrange Interpolation Formula
§ 10 (77–83) The Theorems of S. Bernstein and A. Markov
§ 11 (84–102) Legendre Polynomials and Related Topics
§ 12 (103–113) Further Maximum-Minimum Problems on Polynomials

Determinants and Quadratic Forms

§ 1 (1–16) Evaluation of Determinants. Solution of Linear Equations
§ 2 (17–34) Power Series Expansion of Rational Functions
§ 3 (35–43.2) Generation of Positive Quadratic Forms
§ 4 (44–54.4) Miscellaneous Problems
§ 5 (55–72) Determinants of Systems of Functions

Number Theory

1. Arithmetical Functions
§ 1 (1–11) Problems on the Integral Parts of Numbers
§ 2 (12–20) Counting Lattice Points
§ 3 (21–27.2) The Principle of Inclusion and Exclusion
§ 4 (28–37) Parts and Divisors
§ 5 (38–42) Arithmetical Functions, Power Series, Dirichlet Series
§ 6 (43–64) Multiplicative Arithmetical Functions
§ 7 (65–78) Lambert Series and Related Topics
§ 8 (79–83) Further Problems on Counting Lattice Points

2. Polynomials with Integral Coefficients and Integral-Valued Functions
§ 1 (84–93) Integral Coefficients and Integral-Valued Polynomials
§ 2 (94–115) Integral-Valued Functions and their Prime Divisors
§ 3 (116–129) Irreducibility of Polynomials

3. Arithmetical Aspects of Power Series
§ 1 (130–137) Preparatory Problems on Binomial Coefficients
§ 2 (138–148) On Eisenstein’s Theorem
§ 3 (149–154) On the Proof of Eisenstein’s Theorem
§ 4 (155–164) Power Series with Integral Coefficients Associated with Rational Functions
§ 5 (165–173) Function-Theoretic Aspects of Power Series with Integral Coefficients
§ 6 (174–187) Power Series with Integral Coefficients in the Sense of Hurwitz
§ 7 (188–193) The Values at the Integers of Power Series that Converge about z = ?

4. Some Problems on Algebraic Integers
§ 1 (194–203) Algebraic Integers. Fields
§ 2 (204–220) Greatest Common Divisor
§ 3 (221–227.2) Congruences
§ 4 (228–237) Arithmetical Aspects of Power Series

5. Miscellaneous Problems
§ 1 (237.1–244.4) Lattice Points in Two and Three Dimensions
§ 2 (245–266) Miscellaneous Problems

Geometric Problems

§ 1 (1–25) Some Geometric Problems


§ 1 Additional Problems to Part One

New Problems in English Edition