PART I: ELEMENTS OF THE GENERAL THEORY OF ANALYTIC 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. Behavior 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. Behavior of Analytic Functions at Infinity |
33. The Residue Theorem |
34. Inverses of Analytic Functions |
35. Rational Functions |
Bibliography; Index |
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PART II: APPLICATIONS AND CONTINUATION OF THE GENERAL THEORY |
IntroductionSection I. Single-valued Functions |
Chapter 1. Entire Functions |
1. Weierstrass's Factor-theorem |
2. Proof of Weierstrass's Factor-theorem |
3. Examples of Weierstrass's Factor-theorem |
Chapter 2. Meromorphic Func |
4. Mittag-Leffler's Theorem |
5. Proof of Mittag-Leffler’s Theorem |
6. Examples of Mittag-Leffler's Theorem |
Chapter 3. Periodic Functions |
7. The Periods of Analytic Functions |
8. Simply Periodic Functions |
9. Doubly Periodic Functions; in Particular, Elliptic Functions |
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Section II. Multiple-valued Functions |
Chapter 4. Root and Logarithm |
10. Prefatory Remarks Concerning Multiple-valued Functions and Riemann Surfaces |
11. The Riemann Surfaces for p(root)z and log z |
12. The Riemann Surfaces for the Functions w = root(z – a1)(z – a2) . . . (z – ak) |
Chapter 5. Algebraic Functions |
13. Statement of the Problem |
14. The Analytic Character of the Roots in the Small |
15. The Algebraic Function |
Chapter 6. The Analytic Configuration |
16. The Monogenic Analytic Function |
17. The Riemann Surface |
18. The Analytic Configuration |
Bibliography, Index |