Elements of the Theory of Functions


General background: complex numbers, linear functions, sets and sequences, conformal mapping. Detailed proofs.
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General background: complex numbers, linear functions, sets and sequences, conformal mapping. Detailed proofs.
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Product Details

  • ISBN-13: 9780486601540
  • Publisher: Dover Publications
  • Publication date: 6/1/1952
  • Pages: 140
  • Product dimensions: 5.38 (w) x 7.99 (h) x 0.35 (d)

Table of Contents

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
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