Problems and Theorems in Analysis II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry: Polya, George; Szego, Gabor; Billigheimer, C. E. : 9783540636861: Book: Technology: eCampus.com

From the reviews: "... In the past, more of the leading mathematicians proposed and solved problems than today, and there were problem departments in many journals. PA3lya and Szego must have combed all of the large problem literature from about 1850 to 1925 for their material, and their collection of the best in analysis is a heritage of lasting value. The work is unashamedly dated. With few exceptions, all of its material comes from before 1925. We can judge its vintage by a brief look at the author indices (combined). Let's start on the C's: Cantor, CarathAcodory, Carleman, Carlson, Catalan, Cauchy, Cayley, CesA�ro,... Or the L's: Lacour, Lagrange, Laguerre, Laisant, Lambert, Landau, Laplace, Lasker, Laurent, Lebesgue, Legendre,... Omission is also information: Carlitz, ErdAs, Moser, etc."Bull.Americ.Math.Soc.

Few mathematical books are worth translating 50 years after original publication. PolyA�-SzegA is one! It was published in German in 1924, and its English edition was widely acclaimed when it appeared in 1972. In the past, more of the leading mathematicians proposed and solved problems than today. Their collection of the best in analysis is a heritage of lasting value.

Presents all the large problem literature from about 1850 to 1925 representing the best in analysis of the period. All of the information here is obviously dated but presents a heritage of lasting value.

Part Four. Functions of One Complex Variable. Special Part

Maximum Term and Central Index, Maximum Modulus and Number of Zeros

Analogy between μ(r) and M(r), v(r) and N(r)

(4)

Further Results on μ(r) and v(r)

(1)

Connection between μ(r), v(r), M(r) and N(r)

(3)

μ(r) and M(r) under Special Regularity Assumptions

(3)

Schlicht Mappings

Introductory Material

(1)

Uniqueness Theorems

(1)

Existence of the Mapping Function

(1)

The Inner and the Outer Radius. The Normed Mapping Function

(4)

Relations between the Mappings of Different Domains

(3)

The Koebe Distortion Theorem and Related Topics

(4)

Miscellaneous Problems

Various Propositions

(2)

A Method of E. Landau

(1)

Rectilinear Approach to an Essential Singularity

(1)

Asymptotic Values of Entire Functions

(1)

Further Applications of the Phragmen-Lindelof Method

(1)

Supplementary Problems

(3)

Part Five. The Location of Zeros

Rolle's Theorem and Descartes' Rule of Signs

Zeros of Functions, Changes of Sign of Sequences

(3)

Reversals of Sign of a Function

(1)

First Proof of Descartes' Rule of Signs

(3)

Applications of Descartes' Rule of Signs

(1)

Applications of Rolle's Theorem

(3)

Laguerre's Proof of Descartes' Rule of Signs

(4)

What is the Basis of Descartes' Rule of Signs?

(1)

Generalizations of Rolle's Theorem

(2)

The Geometry of the Complex Plane and the Zeros of Polynomials

Center of Gravity of a System of Points with respect to a Point

(1)

Center of Gravity of a Polynomial with respect to a Point. A Theorem of Laguerre

(3)

Derivative of a Polynomial with respect to a Point. A Theorem of Grace

(5)

Miscellaneous Problems

Approximation of the Zeros of Transcendental Functions by the Zeros of Rational Functions

(4)

Precise Determination of the Number of Zeros by Descartes' Rule of Signs

(3)

Additional Problems on the Zeros of Polynomials

(2)

Part Six. Polynomials and Trigonometric Polynomials

Tchebychev Polynomials

(1)

General Problems on Trigonometric Polynomials

(1)

Some Special Trigonometric Polynomials

(2)

Some Problems on Fourier Series

(2)

Real Non-negative Trigonometric Polynomials

(1)

Real Non-negative Polynomials

(1)

Maximum-Minimum Problems on Trigonometric Polynomials

(2)

Maximum-Minimum Problems on Polynomials

(2)

The Lagrange Interpolation Formula

(2)

The Theorems of S. Bernstein and A. Markov

(1)

Legendre Polynomials and Related Topics

(4)

Further Maximum-Minimum Problems on Polynomials

(3)

Part Seven. Determinants and Quadratic Forms

Evaluation of Determinants. Solution of Linear Equations

(4)

Power Series Expansion of Rational Functions

(4)

Generation of Positive Quadratic Forms

100

(4)

Miscellaneous Problems

104

(4)

Determinants of Systems of Functions

108

(3)

Part Eight. Number Theory

Arithmetical Functions

Problems on the Integral Parts of Numbers

111

(1)

Counting Lattice Points

112

(1)

The Principle of Inclusion and Exclusion

113

(3)

Parts and Divisors

116

(2)

Arithmetical Functions, Power Series, Dirichlet Series

118

(2)

Multiplicative Arithmetical Functions

120

(5)

Lambert Series and Related Topics

125

(3)

Further Problems on Counting Lattice Points

128

(1)

Polynomials with Integral Coefficients and Integral-Valued Functions

Integral Coefficients and Integral-Valued Polynomials

129

(1)

Integral-Valued Functions and their Prime Divisors

130

(2)

Irreducibility of Polynomials

132

(2)

Arithmetical Aspects of Power Series

Preparatory Problems on Binomial Coefficients

134

(1)

On Eisenstein's Theorem

134

(2)

On the Proof of Eisenstein's Theorem

136

(1)

Power Series with Integral Coefficients Associated with Rational Functions

137

(1)

Function-Theoretic Aspects of Power Series with Integral Coefficients

138

(2)

Power Series with Integral Coefficients in the Sense of Hurwitz

140

(2)

The Values at the Integers of Power Series that Converge about z = ∞

142

(1)

Some Problems on Algebraic Integers

Algebraic Integers. Fields

143

(3)

Greatest Common Divisor

146

(1)

Congruences

147

(2)

Arithmetical Aspects of Power Series

149

(1)

Miscellaneous Problems

Lattice Points in Two and Three Dimensions

150

(2)

Miscellaneous Problems

152

(5)

Part Nine. Geometric Problems

Some Geometric Problems

157

(6)

Appendix

§1 Additional Problems to Part One

163

New Problems in English Edition

Author Index

Subject Index

Topics

Errata


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