The present English edition is not a mere translation of the German original. Many new problems have been added and there are also other changes, mostly minor. Yet all the alterations amount to less than ten percent of the text. We intended to keep intact the general plan and the original flavor of the work. Thus we have not introduced any essentially new subject matter, although the mathematical fashion has greatly changed since 1024. We have restricted ours'elves to supplementing the topics originally chosen.

Part Four. Functions of One Complex Variable. Special Part

Chapter 1. Maximum Term and Central Index， Maximum Modulus and Number of Zeros

Problem

Numbers

1 (1-40) Analogy between u(r) and M(r)， v(r) and N (r)

2 (41-47) Further Results on u(r) and v(r)

3 (48-66) Connection between u(r)， v(r)， M(r) and N(r)

4 (67-76) u(r) and M(r) under Special Regularity Assumptions

Chapter 2. Sehlicht 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

Chapter 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）Fulther Applications of the Phragmen-Lindelof Method

6 (206-212) Supplementary Problems

Part Five. The Location of Zeros

Chapter 1. Rolle's Theorem and Descartes' Rule of Signs

Problem

Numbers

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 ;'heorem

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

Chapter 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

Chapter 3. Miscellaneous Problems

1 (157-182) Approximation of the Zeres 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

Part 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 Trigo.nometric 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 Poly-nomials

Part Seven. Determinants and Quadratic Forms

Problem

Numbcrs

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

Part Eight. Number Theory

Chapter 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

Chapter 2. Polynomials with Integral Coefficients and Integral-Valued Functions

1 (84-93) Integral Coefficients and Integral-Valued Poly-nomials

32 (94-115) Integral-Valued Functions and their Prime Divisors

3 (116-129) Irreducibility of Polynomials

Chapter 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 theProofofEisenstein'sTheorem

4 (155-164) Power Series with Integral Coefficients Associated with Rational Functions

5 (165-173) Furlction-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=

Chapter 4. Some Problems on Algebraic Integers

Problem

Numbers

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

Chapter 5. Miscellaneous Problems

1 (237.1-244.4) Lattice Points in Two and Three Dimensions

2 (245-266) Miscellaneous Problems

Part Nine. Geometric Problems

1 (1-25) Some Geometric Problems

Appendix

1 Additional Problems to Part One

New Problems in English Edition

Author Index

Subject Index

Topics

Errata