Theory of Functions Part 1: Elements of the General Theory of Analytical Functions

Preface
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