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## Über dieses Buch

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)

## Inhaltsverzeichnis

### Chapter 1. Operations with Power Series

Without Abstract
George Pólya, Gabor Szegö

### Chapter 2. Linear Transformations of Series. A Theorem of Cesàro

Without Abstract
George Pólya, Gabor Szegö

### Chapter 3. The Structure of Real Sequences and Series

Without Abstract
George Pólya, Gabor Szegö

### Chapter 4. Miscellaneous Problems

Without Abstract
George Pólya, Gabor Szegö

### 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

Without Abstract
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<r1r2r3≦…≦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

Without Abstract
George Pólya, Gabor Szegö

### Chapter 1. Complex Numbers and Number Sequences

Without Abstract
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

Without Abstract
George Pólya, Gabor Szegö

### Chapter 4. Cauchy’s Theorem. The Argument Principle

Without Abstract
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 [251258] 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

### Part One. Infinite Series and Infinite Sequences

Without Abstract
George Pólya, Gabor Szegö

### Part Two. Integration

Without Abstract
George Pólya, Gabor Szegö

### Part Three. Functions of One Complex Variable

Without Abstract
George Pólya, Gabor Szegö

### Backmatter

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