# Book:George Pólya/Problems and Theorems in Analysis I

## George Pólya and Gábor Szegő: Problems and Theorems in Analysis I

Published $1925$, Springer

ISBN 3-540-63640-4 (translated by Dorothée Aeppli).

### Contents

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