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Although the book's title refers only to analysis, the book actually contains a broad range of problems. It starts in [[combinatorics]], and quickly branches out from mathematical analysis to [[number theory]], [[geometry]], [[linear algebra]], and even some [[physics]].<ref name=SzegoCP/>{{rp|23-24}} The specific topics treated bear witness to the special interests of Pólya ([[Descartes' rule of signs]], [[Pólya's enumeration theorem]]), Szegö (polynomials, [[trigonometric polynomials]], and his own work in [[orthogonal polynomials]]) and sometimes both (the zeros of polynomials and [[analytic functions]], [[complex analysis]] in general).<ref name=SzegoCP/>{{rp|25-27}} Many of the book's problems are not new, and their solutions include back-references to their original sources.<ref name=PTA1/>{{rp|xii-xiii, xvii-xviii}} The section on geometry (IX) contains many problems contributed by [[Charles Loewner|Loewner]] (in [[differential geometry]]) and [[Arthur Hirsch|Hirsch]] (in [[algebraic geometry]]).<ref name=SzegoCP/>{{rp|27}}
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 (''[[Mathematics and Plausible Reasoning]]'', ''[[How to Solve It]]'').<ref name=SzegoCP/>{{rp|23-24}} The pair held practice sessions, in which the problems were put to university students and worked through as a class (with some of the representative problems solved by the teacher, and the harder problems set as homework). They went through portions of the book at a rate of about one chapter a semester.<ref name=PTA1/>{{rp|xi-xii}}
A typical example of the progression of questions in ''Problems and Theorems in Analysis'' is given by the first three questions in (the American edition of) volume I:
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