Problems and Theorems in Analysis
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.
"[1]: 59 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.
[3]: 562 [1]: 54 [4]: 11 However Pólya believed their areas of expertise were sufficiently different that the collaboration would prove fruitful.
We were both readers of the same well directed Hungarian Mathematical Journal for high school students that stressed problem solving.
[4]: 11 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.
[1]: 60 Both were also under threat by the rise of antisemitism in Central Europe (Pólya and Szegő were Hungarian Jews).
Financial difficulties, on top of pessimism about appointment to a German university, convinced Pólya to move to England in 1925.
[1]: 61–63 [4]: 23 Szegő took longer to flee, not leaving Germany until 1934 when Pólya and Harald Bohr convinced him to accept a post at Washington University.
By this time the Nazis had already begun purging Jewish professors from German universities.
[6] Szegő and Pólya (who collaborated on little after the problem book) were reunited in America in the 1950s, in the mathematics department of Stanford University.
[1]: 62 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 preface of the book contains some remarks on general problem solving and mathematical heuristics which anticipate Pólya's later works on that subject (Mathematics and Plausible Reasoning, How to Solve It).
[1]: 55 Substantial additions were made in the English translation (published in 1972 and 1976), including new sections and back-references to Pólya's other works on problem solving.
Various eminent mathematicians (Bernays, Courant, Fejér, E. Landau, F. Riesz, Toeplitz) had read over the galley proofs while the work was in press[7]: xii–xiii and its early reviewers (F. Riesz again, Knopp, Tamarkin) were not much less impressive, all effusive in their praise.
[1]: 58–60 The careful pedagogy meant that graduate students were able to learn analysis from Problems and Theorems alone.
