Problems and Theorems in Analysis: Difference between revisions

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'''''Problems and Theorems in Analysis''''' ({{lang-de|Aufgaben und Lehrsätze aus der Analysis|links=no}}) is a two-volume [[problem book]] in [[Mathematical analysis|analysis]] by [[George Pólya]] and [[Gábor Szegő]]. It is highly regarded for the quality of its problems and its method of organisation, not by topic but by method of solution. 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."<ref name=Walks/>{{rp|59}}

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==Background==

[[File:George Polya and Gabor Szego in Berlin.jpg|thumb|Szegő (left) and Polya (right) in [[Berlin]], 1925, delivering the original manuscript of ''Problems and Theorems'' to Springer.<ref>{{cite book |title=The Pólya Picture Album: Encounters of a Mathematician |first=George |last=Polya |date=1987 |publisher=Birkhauser}}</ref>{{rp|63}}]]

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 (at this time, Szegő was only 17, while Pólya was a postdoctoral researcher of 25). Szegő's early career was intertwined with Pólya, his first two papers concerned problems posed by Pólya.<ref name=PolyaObit>{{cite journal |first1=G. L. |last1=Alexanderson |first2=L. H. |last2=Lange |title=Obituary: George Pólya |journal=Bulletin of the London Mathematical Society |volume=19 |issue=6 |date=1987 |pages=559-608559–608 |doi=10.1112/blms/19.6.559 }}</ref>{{rp|562}}<ref name=Walks>{{cite book |last=Alexanderson |first=Gerald L. |title=The Random Walks of George Polya |publisher=The Mathematical Association of America |date=2000 }}</ref>{{rp|54}}<ref name=SzegoCP/>{{rp|11}} However Pólya believed their areas of expertise were sufficiently different that the collaboration would prove fruitful.<ref name=PolyaObit/>{{rp|562}} Pólya and Szegő signed the contract with [[Springer Science+Business Media|Springer-Verlag]] for the book in 1923 and it was published by 1925.<ref name=PolyaMT>{{MacTutor |id=Polya |title=George Pólya |date=November 2002 }}</ref>

Pólya later wrote of the period in which they wrote the book:

{{quoteblockquote|It was a wonderful time; we worked with enthusiasm and concentration. We had similar backgrounds. We were both influenced, like all other young Hungarian mathematicians of that time, by [[Leopold Fejér]]. We were both readers of the same well directed Hungarian Mathematical Journal for high school students that stressed problem solving. We were interested in the same kind of questions, in the same topics; but one of us knew more about one topic and the other more about some other topic. It was a fine collaboration. The book ''Aufgaben und Lehrsatze aus der Analysis'', the result of our cooperation, is my best work and also the best work of Gábor Szegő.<ref name=SzegoCP>{{cite book |title=Collected Papers |volume=1 |last=Szego |first=Gabor |date=1982 |publisher=Birkhäuser |url=https://archive.org/details/collectedpapers0000szeg |url-access=registration }}</ref>{{rp|11}}}}

Writing ''Problems and Theorems'' was an intense experience for both young mathematicians. Pólya's wife worried he might have a nervous breakdown.<ref name=Walks/>{{rp|60}} Both were also under threat by the rise of antisemitism in Germany (both 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.<ref name=Walks/>{{rp|61-6361–63}} 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.<ref name=SzegoMT>{{MacTutor |id=Szego |title=Gábor Szegő |date=July 2014 }}</ref> 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]].<ref name=Walks/>{{rp|62}}

==Reception==