Theory of functions, zeros, polynomials, determinants, number theory, geometry
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Bibliographic Information
Theory of functions, zeros, polynomials, determinants, number theory, geometry
（Classics in mathematics, . Problems and theorems in analysis ; v. 2）
Springer, c1998
Reprint ed.
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Aufgaben und Lehrsätze aus der Analysis
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Note
Reprint of the 1976 edition
Rev. and enl. translation of Aufgaben und Lehrsätze aus der Analysis 2, 4th ed., 1971; Heidelberger Taschenbücher, Bd. 74
"Originally published as v. 216 of the Grundlehren der Mathematischen Wissenschaften" -- T.p. verso
Description and Table of Contents
Description
Few mathematical books are worth translating 50 years after original publication. Polya-Szegoe 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.
Table of Contents
Four. Functions of One Complex Variable. Special Part.- 1. Maximum Term and Central Index, Maximum Modulus and Number of Zeros.- 1 (1-40) Analogy between ?(r) and M(r), v(r) and N(r).- 2 (41-47) Further Results on ?(r) and v(r).- 3 (48-66) Connection between ?(r), v(r), M(r) and N(r).- 4 (67-76) ?(r) and M(r) under Special Regularity Assumptions.- 2. Schlicht Mappings.- 1 (77-83) Introductory Material.- 2 (84-87) Uniqueness Theorems.- 3 (88-96) Existence of the Mapping Function.- 4 (97-120) The Inner and the Outer Radius. The Normed Mapping Function.- 5 (121-135) Relations between the Mappings of Different Domains.- 6 (136-163) The Koebe Distortion Theorem and Related Topics.- 3. Miscellaneous Problems.- 1 (164-174.2) Various Propositions.- 2 (175-179) A Method of E. Landau.- 3 (180-187) Rectilinear Approach to an Essential Singularity.- 4 (188-194) Asymptotic Values of Entire Functions.- 5 (195-205) Further Applications of the Phragmen-Lindeloef Method.- 6 (*206-*212) Supplementary Problems.- Five. The Location of Zeros.- 1. Rolle's Theorem and Descartes' Rule of Signs.- 1 (1-21) Zeros of Functions, Changes of Sign of Sequences.- 2 (22-27) Reversals of Sign of a Function.- 3 (28-41) First Proof of Descartes' Rule of Signs.- 4 (42-52) Applications of Descartes' Rule of Signs.- 5 (53-76) Applications of Rolle's Theorem.- 6 (77-86) Laguerre's Proof of Descartes' Rule of Signs.- 7 (87-91) What is the Basis of Descartes' Rule of Signs?.- 8 (92-100) Generalizations of Rolle's Theorem.- 2. The Geometry of the Complex Plane and the Zeros of Polynomials.- 1 (101-110) Center of Gravity of a System of Points with respect to a Point.- 2 (111-127) Center of Gravity of a Polynomial with respect to a Point. A Theorem of Laguerre.- 3 (128-156) Derivative of a Polynomial with respect to a Point. A Theorem of Grace.- 3. Miscellaneous Problems.- 1 (157-182) Approximation of the Zeros of Transcendental Functions by the Zeros of Rational Functions.- 2 (183-189.3) Precise Determination of the Number of Zeros by Descartes' Rule of Signs.- 3 (190-196.1) Additional Problems on the Zeros of Polynomials.- Six. Polynomials and Trigonometric Polynomials.- 1 (1-7) Tchebychev Polynomials.- 2 (8-15) General Problems on Trigonometric Polynomials.- 3 (16-28) Some Special Trigonometric Polynomials.- 4 (29-38) Some Problems on Fourier Series.- 5 (39-43) Real Non-negative Trigonometric Polynomials.- 6 (44-49) Real Non-negative Polynomials.- 7 (50-61) Maximum-Minimum Problems on Trigonometric Polynomials.- 8 (62-66) Maximum-Minimum Problems on Polynomials.- 9 (67-76) The Lagrange Interpolation Formula.- 10 (77-83) The Theorems of S. Bernstein and A. Markov.- 11 (84-102) Legendre Polynomials and Related Topics.- 12 (103-113) Further Maximum-Minimum Problems on Polynomials.- Seven. Determinants and Quadratic Forms.- 1 (1-16) Evaluation of Determinants. Solution of Linear Equations.- 2 (17-34) Power Series Expansion of Rational Functions.- 3 (35-43.2) Generation of Positive Quadratic Forms.- 4 (44-54.4) Miscellaneous Problems.- 5 (55-72) Determinants of Systems of Functions.- Eight. Number Theory.- 1. Arithmetical Functions.- 1 (1-11) Problems on the Integral Parts of Numbers.- 2 (12-20) Counting Lattice Points.- 3 (21-27.2) The Principle of Inclusion and Exclusion.- 4 (28-37) Parts and Divisors.- 5 (38-42) Arithmetical Functions, Power Series, Dirichlet Series.- 6 (43-64) Multiplicative Arithmetical Functions.- 7 (65-78) Lambert Series and Related Topics.- 8 (79-83) Further Problems on Counting Lattice Points.- 2. Polynomials with Integral Coefficients and Integral-Valued Functions.- 1 (84-93) Integral Coefficients and Integral-Valued Polynomials.- 2 (94-115) Integral-Valued Functions and their Prime Divisors.- 3 (116-129) Irreducibility of Polynomials.- 3. Arithmetical Aspects of Power Series.- 1 (130-137) Preparatory Problems on Binomial Coefficients.- 2 (138-148) On Eisenstein's Theorem.- 3 (149-154) On the Proof of Eisenstein's Theorem.- 4 (155-164) Power Series with Integral Coefficients Associated with Rational Functions.- 5 (165-173) Function-Theoretic Aspects of Power Series with Integral Coefficients.- 6 (174-187) Power Series with Integral Coefficients in the Sense of Hurwitz.- 7 (188-193) The Values at the Integers of Power Series that Converge about z = ?.- 4. Some Problems on Algebraic Integers.- 1 (194-203) Algebraic Integers. Fields.- 2 (204-220) Greatest Common Divisor.- 3 (221-227.2) Congruences.- 4 (228-237) Arithmetical Aspects of Power Series.- 5. Miscellaneous Problems.- 1 (237.1-244.4) Lattice Points in Two and Three Dimensions.- 2 (245-266) Miscellaneous Problems.- Nine. Geometric Problems.- 1 (1-25) Some Geometric Problems.- Errata.- 1 Additional Problems to Part One.- New Problems in English Edition.- Author Index.- Topics.
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