**Konrad Knopp**. The first is Infinite

*Sequences and Series*which contains the following publisher's information:-

One of the finest expositors in the field of modern mathematics, Dr Konrad Knopp here concentrates on a topic that is of particular interest to20th-century mathematicians and students. He develops the theory of infinite sequences and series from its beginnings to a point where the reader will be in a position to investigate more advanced stages on his own. The foundations of the theory are therefore presented with special care, while the developmental aspects are limited by the scope and purpose of the book.

All definitions are clearly stated; all theorems are proved with enough detail to make them readily comprehensible. The author begins with the construction of the system of real and complex numbers, covering such fundamental concepts as sets of numbers abd functions of real and complex variables. In the treatment of sequences and series that follows, he covers arbitrary and null sequences; sequences and sets of numbers; convergence and divergence; Cauchy's limit theorem; main tests for sequences; and infinite series. Chapter three deals with main tests for infinite series and operating with convergent series. Chapters four and five explain power series and the development of the theory of convergence, while chapter six treats expansion of the elementary functions. The book concludes with a discussion of numerical and closed evaluation of series.

### Here is the list of chapter headings of the text:

**Foreword**

**Chapter 1. Introduction and Prerequisites**

1.1 Preliminary remarks concerning sequences and series

1.2 Real and complex numbers

1.3 Sets of numbers

1.4 Functions of a real and of a complex variable

**Chapter 2. Sequences and Series**

2.1 Arbitrary sequences. Null sequences

2.2 Sequences and sets of numbers

2.3 Convergence and divergence

2.4 Cauchy's limit theorem and its generalizations

2.5 The main tests for sequences

2.6 Infinite series

**Chapter 3. The Main Tests for Infinite Series. Operating with Convergent Series**

3.1 Series of positive terms: The first main test and the comparison tests of the first and second kind

3.2 The radical test and the ratio test

3.3 Series of positive, monotonically decreasing terms

3.4 The second main test

3.5 Absolute convergence

3.6 Operating with convergent series

3.7 Infinite products

**Chapter 4. Power Series**

4.1 The circle of convergence

4.2 The functions represented by power series

4.3 Operating with power series. Expansion of composite functions

4.4 The inversion of a power series

**Chapter 5. Development of the Theory of Convergence**

5.1 The theorems of Abel, Dini, and Pringsheim

5.2 Scales of convergence tests

5.3 Abel's partial summation. Lemmas

5.4 Special comparison tests of the second kind

5.5 Abel's and Dirichlet's tests and their generalizations

5.6 Series transformations

5.7 Multiplication of series

**Chapter 6. Expansion of the Elementary Functions**

6.1 List of the elementary functions

6.2 The rational functions

6.3 The exponential function and the circular functions

6.4 The logarithmic function

6.5 The general power and the binomial series

6.6 The cyclometric functions

**Chapter 7. Numerical and Closed Evaluation of Series**

7.1 Statement of the problem

7.2 Numerical evaluations and estimations of remainders

7.3 Closed evaluations

**Bibliography**

**Index**

Next we give the contents of Knopp's famous book

*Theory of Functions.*

**Section I. Fundamental Concepts**

**Chapter 1. Numbers and Points**

1. Prerequisites

2. The Plane and Sphere of Complex Numbers

3. Point Sets and Sets of Numbers

4. Paths, Regions, Continua

**Chapter 2. Functions of a Complex Variable**

5. The Concept of a Most General (Single-valued) Function of a Complex Variable

6. Continuity and Differentiability

7. The Cauchy-Riemann Differential Equations

**Section II. Integral Theorems**

**Chapter 3. The Integral of a Continuous Function**

8. Definition of the Definite Integral

9. Existence Theorem for the Definite Integral

10. Evaluation of Definite Integrals

11. Elementary Integral Theorems

**Chapter 4. Cauchy's Integral Theorem**

12. Formulation of the Theorem

13. Proof of the Fundamental Theorem

14. Simple Consequences and Extensions

**Chapter 5. Cauchy's Integral Formulas**

15. The Fundamental Formula

16. Integral Formulas for the Derivatives

**Section III. Series and the Expansion of Analytic Functions in Series**

**Chapter 6. Series with Variable Terms**

17. Domain of Convergence

18. Uniform Convergence

19. Uniformly Convergent Series of Analytic Functions

**Chapter 7. The Expansion of Analytic Functions in Power Series**

20. Expansion and Identity Theorems for Power Series

21. The Identity Theorem for Analytic Functions

**Chapter 8. Analytic Continuation and Complete Definition of Analytic Functions**

22. The Principle of Analytic Continuation

23. The Elementary Functions

24. Continuation by Means of Power Series and Complete Definition of Analytic Functions

25. The Monodromy Theorem

26. Examples of Multiple-valued Functions

**Chapter 9. Entire Transcendental Functions**

27. Definitions

28. Behaviour for Large | *z* |

**Section IV. Singularities**

**Chapter 10. The Laurent Expansion**

29. The Expansion

30. Remarks and Examples

**Chapter 11. The Various types of Singularities**

31. Essential and Non-essential Singularities or Poles

32. Behaviour of Analytic Functions at Infinity

33. The Residue Theorem

34. Inverses of Analytic Functions

35. Rational Functions

**Bibliography**

**Index**

*Problem Book in the Theory of Functions:-*

Volume I contains more than300elementary problems dealing with fundamental concepts, infinite sequences and series, functions of a complex variable, conformal mapping, and more.

Volume II features over230problems in advanced theory - singularities, entire and meromorphic functions, periodic functions, analytic continuation, multiple-valued functions and Riemann surfaces, and more. Solution hints are provided, plus complete solutions to all problems.

**Chapter I. Fundamental Concepts**

1. Numbers and Points. Problems; Answers

2. Point Sets. Paths. Regions. Problems; Answers

**Chapter II. Infinite Sequences and Series**

3. Limits of Sequences. Infinite Series with Constant Terms. Problems; Answers

4. Convergence Properties of Power Series. Problems; Answers

**Chapter III. Functions of a Complex Variable**

5. Limits of Functions. Continuity and Differentiability. Problems; Answers

6. Simple Properties of the Elementary Functions. Problems; Answers

**Chapter IV. Integral Theorems**

7. Integration in the Complex Domain. Problems; Answers

8. Cauchy's Integral Theorems and Integral Formulas. Problems; Answers

**Chapter V. Expansion in Series**

9. Series with Variable Terms. Uniform Convergence. Problems; Answers

10. Expansion in Power Series. Problems; Answers

11. Behaviour of Power Series on the Circle of Convergence. Problems; Answers

**Chapter V. Conformal Mapping**

12. Linear Functions. Stereographic Projection. Problems; Answers

13. Simple Non-Linear Mapping Problems. Problems; Answers