Konrad Knopp: Texts
One of the finest expositors in the field of modern mathematics, Dr Konrad Knopp here concentrates on a topic that is of particular interest to 20th-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. |
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
Volume I contains more than 300 elementary problems dealing with fundamental concepts, infinite sequences and series, functions of a complex variable, conformal mapping, and more. |
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
JOC/EFR August 2006
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