Hjem

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 Phragmén-Lindelöf 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.