# Functions of a Complex Variable and Some of Their Applications

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Functions of a Complex Variable and Some of Their Applications, Volume 1, discusses the fundamental ideas of the theory of functions of a complex variable. The book is the result of a complete rewriting and revision of a translation of the second (1957) Russian edition. Numerous changes and additions have been made, both in the text and in the solutions of the Exercises. The book begins with a review of arithmetical operations with complex numbers. Separate chapters discuss the fundamentals of complex analysis; the concept of conformal transformations; the most important of the elementary functions; and the complex potential for a plane vector field and the application of the simplest methods of function theory to the analysis of such a field. Subsequent chapters cover the fundamental apparatus of the theory of regular functions, i.e. basic integral theorems and expansions in series; the general concept of an analytic function; applications of the theory of residues; and polygonal domain mapping. This book is intended for undergraduate and postgraduate students of higher technical institutes and for engineers wishing to increase their knowledge of theory.

## Table of Contents

From the Foreword to the First Edition

Foreword to the English Edition

Introduction

1. Complex Numbers

2. The Simplest Operations

3. Multiplication, Division, Integral Powers and Roots

4. Complex Powers. Logarithms

Exercises

I. The Fundamental Ideas of Complex Analysis

5. The Sphere of Complex Numbers

6. Domains and Their Boundaries

7. The Limit of a Sequence

8. Complex Functions of a Real Variable

9. The Complex Form of an Oscillation

10. Functions of a Complex Variable

11. Examples

12. The Limit of a Function

13. Continuity

14. The Cauchy-Riemann Conditions

Exercises

II. Conformal Mappings

15. Conformal Mappings

16. Conformal Mapping of Domains

17. Geometric Significance of the Differential Dw

18. Bilinear Mappings

19. The Circle Property

20. Invariance of the Conjugate Points

21. Conditions Determining Bilinear Mappings

22. Particular Examples

23. General Principles of the Theory of Conformal Mapping

Exercises

III. Elementary Functions

24. The Functions W=Zn and Their Riemann Surfaces

25. The Concept of a Regular Branch. The Functions W=N√2

26. The Function W = 1/2(Z+Z-L) and Its Riemann Surface

27. Examples

28. The Joukowski Profile

29. The Exponential Function and Its Riemann Surface

30. The Logarithmic Function

31. Trigonometrical and Hyperbolic Functions

32. The General Power

33. Examples

Exercises

IV. Applications to the Theory of Plane Fields

34. Plane Vector Fields

35. Examples of Plane Fields

36. Properties of Plane Vector Fields

37. The Force Function and Potential Function

38. The Complex Potential In Electrostatics

39. The Complex Potential In Hydrodynamics and Heat Conduction

40. The Method of Conformal Mapping

41. The Field In a Strip

42. The Field in a Ring Domain

43. Streamlining an Infinite Curve

44. The Problem of Complete Streamlining. Chaplygin's Condition

45. Other Methods

Exercises

V. The Integral Representation of a Regular Function. Harmonic Functions

46. The Integral of a Function of a Complex Variable

47. Cauchy's Integral Theorem

48. Cauchy's Residue Theorem. Chaplygin's Formula

49. The Indefinite Integral

50. Integration of Powers of (Z—A)

51. Cauchy's Integral Formula

52. The Existence of Higher Derivatives

53. Properties of Regular Functions

54. Harmonic Functions

55. Dirichlet's Problem

56. The Integrals of Poisson and Schwarz

57. Applications to the Theory of Plane Fields

Exercises

VI. Representation of Regular Functions By Series

58. Series In the Complex Domain

59. Weierstrass's Theorem

60. Power Series

61. Representation of Regular Functions by Taylor Series

62. The Zeros of a Regular Function. The Uniqueness Theorem

63. Analytic Continuation. Analytic Functions

64. Laurent Series

65. Isolated Singularities

66. Removable Singularities

67. Poles

68. Essential Singularities

69. Behaviour of a Function at Infinity

70. Joukowski's Theorem on the Thrust on an Aerofoil

71. The Simplest Classes of Analytic Functions

Exercises

VII. Applications of the Theory of Residues

72. Evaluation of Integrals of the Form 2π∫0 R(Sin X, Cos X) . Dx

73. Integrals of the Form +∞∫-∞ R(X) . {Sin Cos}Xx . Dx

74. Other Integrals 296

75. Integrals Involving Multi-Valued Functions

76. The Representation of Functions by Integrals

77. The Logarithmic Residue

78. Expansion of Cot Z in Simple Fractions. Mittag-Leffler's Theorem

79. Expansion of Sin Z as an Infinite Product. Weierstrass's Theorem

80. Euler's Gamma Function Γ(Z)

81. Integral Representations of the Γ-Function

Exercises

VIII. Mapping of Polygonal Domains

82. The Symmetry Principle

83. Illustrative Examples

84. The Schwarz-Christoifel Integral

85. Degenerate Cases

86. Illustrative Examples

87. Determination of the Field at the Edges of a Condenser. Rogowski's Condenser

88. The Field of Angular Electrodes

89. The Mapping of Rectangular Domains. Introduction to Elliptic Integrals

90. Introduction to Jacobian Elliptic Functions

Exercises

Answers and Hints for Solution of Exercises

Index

## Product details

- No. of pages: 458
- Language: English
- Copyright: © Pergamon 1964
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483155050

## About the Authors

### B. A. Fuchs

### B. V. Shabat

## About the Editors

### I. N. Sneddon

### S. Ulam

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