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1998 | Book

Operations with Power Series Read chapter Read first chapter

Problems and Theorems in Analysis I

Series. Integral Calculus. Theory of Functions

Authors: George Pólya, Gabor Szegö

Publisher: Springer Berlin Heidelberg

Book Series : Grundlehren der mathematischen Wissenschaften

Included in: Professional Book Archive

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About this book

From the reviews: "The work is one of the real classics of this century; it has had much influence on teaching, on research in several branches of hard analysis, particularly complex function theory, and it has been an essential indispensable source book for those seriously interested in mathematical problems. These volumes contain many extraordinary problems and sequences of problems, mostly from some time past, well worth attention today and tomorrow. Written in the early twenties by two young mathematicians of outstanding talent, taste, breadth, perception, perseverence, and pedagogical skill, this work broke new ground in the teaching of mathematics and how to do mathematical research. (Bulletin of the American Mathematical Society)

Table of Contents

Frontmatter

Infinite Series and Infinite Sequences

Chapter 1. Operations with Power Series
George Pólya, Gabor Szegö
Chapter 2. Linear Transformations of Series. A Theorem of Cesàro
George Pólya, Gabor Szegö
Chapter 3. The Structure of Real Sequences and Series
George Pólya, Gabor Szegö
Chapter 4. Miscellaneous Problems
George Pólya, Gabor Szegö

Integration

Chapter 1. The Integral as the Limit of a Sum of Rectangles
Abstract
Let f(x) be a bounded function on the finite interval [a, b]. The points with abscissae x0, x1, x2,…,xn−1, xn where
$$a = {x_0} < {x_1} < {x_2} < \cdots < {x_{n - 1}} < {x_n} = b,$$
constitute a subdivision of this interval. Denote by m v and M v the greatest lower and the least upper bound, respectively, of f(x) on the v-th subinterval [x v −1, x v ], v= 1,2,…, n. We call
$$L = \sum\limits_{v = 1}^n {{m_v}} ({x_v} - {x_{v - 1}})\,{\text{the }}lower sum{\text{,}}$$
$$U = \sum\limits_{v = 1}^n {{M_v}} ({x_v} - {x_{v - 1}}){\text{the }}upper\,sum$$
belonging to the subdivision x0, x1, x2,…, xn−1, x n . Any upper sum is always larger (not smaller) than any lower sum, regardless of the subdivision considered.
George Pólya, Gabor Szegö
Chapter 2. Inequalities
Abstract
Let a1, a2,…,a n be arbitrary real numbers. Their arithmetic mean U(a) is defined as the expression
$$\mathfrak{A}\left( a \right) = \frac{{{a_1} + {a_2} + \cdots + {a_n}}}{n}.$$
George Pólya, Gabor Szegö
Chapter 3. Some Properties of Real Functions
George Pólya, Gabor Szegö
Chapter 4. Various Types of Equidistribution
Abstract
In the sequel we are considering monotone sequences of positive numbers. The counting function N(r) of such a sequence r1, r2,…, r n ,…, 0<r1≦r2≦r3≦…≦r n ≦…, is defined as the number of those r n ’s that are not larger than r, r ≧ 0:
$$N\left( r \right) = \sum\limits_{{r_n} \leqq r} {1.} $$
George Pólya, Gabor Szegö
Chapter 5. Functions of Large Numbers
George Pólya, Gabor Szegö

Functions of One Complex Variable General Part

Chapter 1. Complex Numbers and Number Sequences
George Pólya, Gabor Szegö
Chapter 2. Mappings and Vector Fields
Abstract
If we associate each point z of some domain D of the z-plane with a certain complex value w according to a given law then w is called a function of z. Two geometrical interpretations of the functional relation are particularly useful. One uses one plane, the other two planes. The value w belonging to the point z (or, if more expedient, \(\bar w \) can be thought of as a vector acting on the point z; in this way a vector field is defined in the domain D. In the other interpretation, the value w associated with the point z in the z-plane is conceived as a point in another complex plane (w-plane). In this way the domain D is mapped onto a certain point set of the w-plane.
George Pólya, Gabor Szegö
Chapter 3. Some Geometrical Aspects of Complex Variables
George Pólya, Gabor Szegö
Chapter 4. Cauchy’s Theorem. The Argument Principle
George Pólya, Gabor Szegö
Chapter 5. Sequences of Analytic Functions
Abstract
The power series
$${a_1}z + {a_2}{z^2} + \cdots + {a_n}{z^n} + \cdots = w$$
which converges not only for z = 0 and for which a1≠ 0 establishes a conformal one to one mapping of a certain neighbourhood of z = 0 onto a certain neighbourhood of w = 0. Consequently the relationship between z and w can also be represented by the expansion
$${b_1}w + {b_2}{w^2} + \cdots + {b_n}{w^n} + \cdots = z,$$
a1b1 = 1. To compute the second series from the first we set
$$\frac{1}{{{a_1} + {a_2}z + {a_3}{z^2} + \cdots + {a_n}{z^{n - 1}} + \cdots }} = \varphi \left( z \right).$$
George Pólya, Gabor Szegö
Chapter 6. The Maximum Principle
Abstract
The values that an analytic function assumes in the different parts of its domain of existence are related to each other: they are connected by analytic continuation and it is impossible to modify the values in one part without inducing a change throughout. Therefore an analytic function can be compared to an organism the main characteristic of which is exactly this: Action on any part calls forth a reaction of the entire system. E.g. the propagation of convergence [251–258] can be compared to the spreading of an infection. Mr. Borel advanced ingenious reflections upon similar comparisons1. We shall examine in what manner the moduli of the values are related that the function assumes in different parts of its domain of existence.
George Pólya, Gabor Szegö

Solutions

Part One. Infinite Series and Infinite Sequences
George Pólya, Gabor Szegö
Part Two. Integration
George Pólya, Gabor Szegö
Part Three. Functions of One Complex Variable
George Pólya, Gabor Szegö
Backmatter
Metadata
Title
Problems and Theorems in Analysis I
Authors
George Pólya
Gabor Szegö
Copyright Year
1998
Publisher
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-61983-0
Print ISBN
978-3-540-63640-3
DOI
https://doi.org/10.1007/978-3-642-61983-0