Problems and Theorems in Analysis
Problems and Theorems in Analysis is a two-volume problem book in analysis by George Pólya and Gábor Szegő. Published in 1925, the two volumes are titled Series. Integral Calculus. Theory of Functions.; and Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry.
The volumes are highly regarded for the quality of their problems and their method of organisation, not by topic but by method of solution, with a focus on cultivating the student's problem-solving skills. Each volume contains problems at the beginning and solutions at the end. As two authors have put it, "there is a general consensus among mathematicians that the two-volume Pólya-Szegő is the best written and most useful problem book in the history of mathematics."
Background
It was Pólya who had the idea for a comprehensive problem book in analysis first, but he realised he would not be able complete it alone. He decided to write it with Szegő, who had been a friend of Pólya's since 1913, when the pair met in Budapest. Szegő's early career was intertwined with Pólya, his first two papers concerned problems posed by Pólya. However Pólya believed their areas of expertise were sufficiently different that the collaboration would prove fruitful. Pólya and Szegő signed the contract with Springer-Verlag for the book in 1923 and it was published by 1925.Pólya later wrote of the period in which they wrote the book:
Writing Problems and Theorems was an intense experience for both young mathematicians. Pólya was a professor in Zurich and Szegő was a Privatdozent in Berlin, so both had independent workloads. Pólya's wife worried he might have a nervous breakdown. Both were also under threat by the rise of antisemitism in Central Europe. Financial difficulties, on top of pessimism about appointment to a German university, convinced Pólya to move to England in 1925. Szegő took longer to flee, not leaving Germany until 1934 when Pólya and Harald Bohr convinced him to accept a post at Washington University in St. Louis. By this time the Nazis had already begun purging Jewish professors from German universities. Szegő and Pólya were reunited in America in the 1950s, in the mathematics department of Stanford University.
Contents
Although the book's title refers only to analysis, a broad range of problems are contained within. It starts in combinatorics, and quickly branches out from mathematical analysis to number theory, geometry, linear algebra, and even some physics. The specific topics treated bear witness to the special interests of Pólya, Szegö and sometimes both. Many of the book's problems are not new, and their solutions include back-references to their original sources. The section on geometry contains many problems contributed by Loewner and Hirsch.The book was unique at the time because of its arrangement, less by topic and more by method of solution, so arranged in order to build up the student's problem-solving abilities. The preface of the book contains some remarks on general problem solving and mathematical heuristics which anticipate Pólya's later works on that subject. The pair held practice sessions, in which the problems were put to university students and worked through as a class. They went through portions of the book at a rate of about one chapter a semester.
A typical example of the progression of questions in Problems and Theorems in Analysis is given by the first three questions in volume I:
1. In how many different ways can you change one dollar? That is, in how many different ways can you pay 100 cents using five different kinds of coins, cents, nickels, dimes, quarters and half-dollars ?
2. Let n stand for a non-negative integer and let denote the number of solutions of the Diophantine equation
in non-negative integers. Then the series
represents a rational function of. Find it.
3. In how many ways can you put the necessary stamps in one row on an airmail letter sent inside the U.S., using 2, 4, 6, 8 cent stamps? The postage is 10 cents.
The first question sets up an elementary combinatorics question; but the second suggests both a solution and a generalisation. The third gives another combinatorics question which can be solved by means of generating functions. Indeed, questions 1-26 follow generating function through further examples. Whole areas of mathematics are developed in this way.
Substantial additions were made in the English translation, including new sections and back-references to Pólya's other works on problem solving.
Reception
and Paul Nevai wrote of the book that, "there is a general consensus among mathematicians that the two-volume Pólya-Szegő is the best written and most useful problem book in the history of mathematics." The book has had its admirers. Various eminent mathematicians had read over the galley proofs while the work was in press and its early reviewers, not much less eminent, were all effusive in their praise. The careful pedagogy meant that graduate students were able to learn analysis from Problems and Theorems alone. Paul Erdős once approached a young mathematician with a problem taken from volume II and announced "I will give $10 to China if you can solve this problem in ten minutes".A Russian translation was published in 1937–38. An English translation was published in 1972–76.