**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 μ(r) and M(r), ν(r) and N(r)

§ 2 (41–47) Further Results on μ(r) and ν(r)

§ 3 (48–66) Connection between μ(r), ν(r), M(r) and N(r)

§ 4 (67–76) μ(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