Functions of a Complex Variable and Some of Their Applications
- 1st Edition - September 13, 2014
- Latest edition
- Authors: B. A. Fuchs, B. V. Shabat
- Editors: I. N. Sneddon, S. Ulam
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 2 2 6 3 - 2
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 5 0 5 - 0
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… Read more
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.
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
- Edition: 1
- Latest edition
- Published: September 13, 2014
- Language: English
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