Two volumes of a classic 5-volume work in one handy edition. Part I considers general foundations of the theory of functions; Part II stresses special functions and characteristic, important types of functions, selected from single-valued and multiple-valued classes. Demonstrations are full and proofs given in detail. Introduction. Bibliographies.

Table of Contents for Theory of Functions, Parts I and II


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

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

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

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