Elements of the Theory of Functions

Section I. Complex Numbers and their Geometric Representation
Chapter I. Foundations
1. Introduction
2. The system of real numbers
3. Pointgs and vectors of the plane
Chapter II. The System of Complex Numbers and the Gaussian Plane of Numbers
4. Historical remarks
5. Introduction of complex numbers. Notation
6. Equality and inequality
7. Addition and subtraction
8. Multiplication and division
9. Derived rules. Powers
10. The system of complex numbers as an extension of the system of real numbers
11. Trigonometric representation of complex numbers
12. Geometric representation of multiplication and division
13. Inequalities and absolute values. Examples
Chapter III. The Riemann Sphere of Numbers
14. The stereographic projection
15. The Riemann sphere of numbers. The point infinity. Examples
Section II. Linear Functions and Circular Transformations
Chapter IV. Mapping by Means of Linear Functions
16. Mapping by means of entire linear functions
17. Mapping by means of the function w = 1/z
18. Mapping by means of arbitrary linear functions
Chapter V. Normal Forms and Particular Linear Mappings
19. The group-property of linear transformations
20. Fixed points and normal forms
21. Particular linear mappings. Cross ratios
22. Further examples
Section III. Sets and Sequences. Power Series
Chapter VI. Point Sets and Sets of Numbers
23. Point sets
24. Sets of real numbers
25. The Bolzano-Weierstrass theorem
Chapter VII. Sequences of Numbers. Infinite Series
26. Sequences of complex numbers
27. Sequences of real numbers
28. Infinite series
Chapter VIII. Power Series
29. The circle of convergence
30. Operations on power series
Section IV. Analytic Functions and Conformal Mapping
Chapter IX. Functions of a Complex Variable
31. The concept of a function of a complex variable
32. Limits of functions
33. Continuity
34. Differentiability
35. Properties of functions represented by power series
Chapter X. Analytic Functions and Conformal Mapping
36. Analytic functions
37. Conformal mapping
Section V. The Elementary Functions
Chapter XI. Power and Root. The Rational Functions
38. Power and root
39. The entire rational functions
40. The fractional rational functions
Chapter XII. The Exponential, Trigonometric, and Hyperbolic Functions
41. The exponential function
42. The functions cos z and sin z
43. The functions tan z and cot z
44. The hyperbolic functions
Chapter XIII. The Logarithm, the Cyclometric Functions, and the Binomial Series
45. The logarithm
46. The cyclometric functions
47. The binomial series and the general power
Bibliography; Index