Problems and Theorems in Analysis II

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.