Problems and Theorems in Analysis II: Theory of Functions. Zeros. Polynomials. Determinants. Number Theory. Geometry (Paperback)

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-Lindelof 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) O