Book:George Pólya/Problems and Theorems in Analysis I

Subject Matter

 * Preface to the English Edition (Stanford, March 1972)


 * Preface to the First German Edition (Zurich and Berlin, October 1924)


 * Part One: Infinite Series and Infinite Sequences
 * Chapter 1: Operations with Power Series
 * $$\S 1$$ (1 - 31). Additive Number Theory, Combinatorial Problems and Applications
 * $$\S 2$$ (31. 1 - 43.1). Binomial Coefficients and Related Problems
 * $$\S 3$$ (44 - 49). Differentiation of Power Series
 * $$\S 4$$ (50 - 60). Functional Equations and Power Series
 * $$\S 5$$ (60.1 - 60.11). Gaussian Binomial Coefficients
 * $$\S 6$$ (61 - 64.2). Majorant Series


 * Chapter 2 Linear Transformations of Series. A Theorem of Cesàro
 * $$\S 1$$ (65 - 78) Triangular Transformations of Sequences into Sequences
 * $$\S 2$$ (79 - 82) More General Transformations of Sequences into Sequences
 * $$\S 3$$ (83 - 97) Transformations of Sequences into Functions. Theorem of Cesàro


 * Chapter 3 The Structure of Real Sequences and Series
 * $$\S 1$$ (98 - 112). The Structure of Infinite Sequences
 * $$\S 2$$ (113 - 116). Convergence Exponent
 * $$\S 3$$ (117 - 123). The Maximum Term of a Power Series
 * $$\S 4$$ (124 - 132). Subseries
 * $$\S 5$$ (132.1 - 137). Rearrangement of the Terms
 * $$\S 6$$ (138 - 13). Distribution of the Signs of the Terms


 * Chapter 4 Miscellaneous Problems
 * $$\S 1$$ (140 - 155). Enveloping Series
 * $$\S 2$$ (156 - 185.2). Various Propositions on Real Series and Sequences
 * $$\S 3$$ (186 - 210). Partitions of Sets, Cycles in Permutations


 * Part Two: Integration


 * Chapter 1 The Integral as the Limit of a Sum of Rectangles
 * $$\S 1$$ (1 - 7). The Lower and the Upper Sum
 * $$\S 2$$ (8 - 19.2). The Degree of Approximation
 * $$\S 3$$ (20 - 29). Improper Integrals Between Finite Limits
 * $$\S 4$$ (30 - 40). Improper Integrals Between Infinite Limits
 * $$\S 5$$ (41 - 47). Applications to Number Theory
 * $$\S 6$$ (48 - 59). Mean Values and Limits of Products
 * $$\S 7$$ (60 - 68). Multiple Integrals


 * Chapter 2 Inequalities
 * $$\S 1$$ (69 - 94). Inequalities
 * $$\S 2$$ (94.1 - 97). Some Applications of Inequalities


 * Chapter 3 Some Properties on Real Functions
 * $$\S 1$$ (98 - 111). Proper Integrals
 * $$\S 2$$ (112 - 118.1). Improper Integrals
 * $$\S 3$$ (119 - 127). Continuous, Differentiable, Convex Functions
 * $$\S 4$$ (128 - 146). Singular Integrals. Weierstrass' Approximation Theorem


 * Chapter 4 Various Types of Equidistribution
 * $$\S 1$$ (147 - 161). Counting Function. Regular Sequences
 * $$\S 2$$ (162 - 165). Criteria of Equidistribution
 * $$\S 3$$ (166 - 173). Multiples of an Irrational Number
 * $$\S 4$$ (174 - 184). Distribution of the Digits in a Table of Logarithms and Related Questions
 * $$\S 5$$ (185 - 194). Other Types of Equidistribution


 * Chapter 5 Functions of Large Numbers
 * $$\S 1$$ (195 - 209). Laplace's Method
 * $$\S 2$$ (210 - 217.1). Modifications of the Method
 * $$\S 3$$ (218 - 222). Asymptotic Evaluation of Some Maxima
 * $$\S 4$$ (223 - 226). Minimax and Maximin


 * Part Three: Functions of One Complex Variable. General Part


 * Chapter 1 Complex Numbers and Number Sequences
 * $$\S 1$$ (1 - 15). Regions and Curves. Working with Complex Variables
 * $$\S 2$$ (16 - 27). Location of the Roots of Algebraic Equations
 * $$\S 3$$ (28 - 35). Zeros of Polynomials, Continued. A Theorem of Gauss
 * $$\S 4$$ (36 - 43). Sequences of Complex Numbers
 * $$\S 5$$ (44 - 50). Sequences of Complex Numbers, Continued: Transformation of Sequences
 * $$\S 6$$ (51 - 54). Rearrangement of Infinite Series


 * Chapter 2 Mappings and Vector Fields
 * $$\S 1$$ (55 - 59). The Cauchy-Riemann Differential Equations
 * $$\S 2$$ (60 - 84). Some Particular Elementary Mappings
 * $$\S 3$$ (85 - 102). Vector Fields


 * Chapter 3 Some Geometrical Aspects of Complex Variables
 * $$\S 1$$ (103 - 116). Mappings of the Circle. Curvature and Support function
 * $$\S 2$$ (117 - 123). Mean Values Along a Circle
 * $$\S 3$$ (124 - 129). Mappings of the Disk. Area
 * $$\S 4$$ (130 - 144). The Modular Graph. The Maximum Principle


 * Chapter 4 Cauchy's Theorem. The Argument Principle
 * $$\S 1$$ (145 - 171). Cauchy's Formula
 * $$\S 2$$ (172 - 178). Poisson's and Jensen's Formulas
 * $$\S 3$$ (179 - 193). The Argument Principle
 * $$\S 4$$ (194 - 206.2). Rouché's Theorem


 * Chapter 5 Sequences of Analytic Functions
 * $$\S 1$$ (207 - 229). Lagrange's Series. Applications
 * $$\S 2$$ (230 - 240). The Real Part of a Power Series
 * $$\S 3$$ (241 - 247). Poles on the Circle of Convergence
 * $$\S 4$$ (248 - 250). Identically Vanishing Power Series
 * $$\S 5$$ (251 - 258). Propagation of Convergence
 * $$\S 6$$ (259 - 262). Convergence in Separated Regions
 * $$\S 7$$ (263 - 265). The Order of Growth of Certain Sequences of Polynomials


 * Chapter 6 The Maximum Principle
 * $$\S 1$$ (266 - 279). The Maximum Principle of Analytic Functions
 * $$\S 2$$ (280 - 298). Schwarz's Lemma
 * $$\S 3$$ (299 - 310). Hadamard's Three Circle Theorem
 * $$\S 4$$ (311 - 321). Harmonic Functions
 * $$\S 5$$ (322 - 340). The Phragmén-Lindelöf Method


 * Author Index


 * Subject Index