Functions of a Complex Variable and Some of Their Applications

Functions of a Complex Variable and Some of Their Applications

1st Edition - January 1, 1964

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  • Authors: B. A. Fuchs, B. V. Shabat
  • eBook ISBN: 9781483155050

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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


    1. Complex Numbers

    2. The Simplest Operations

    3. Multiplication, Division, Integral Powers and Roots

    4. Complex Powers. Logarithms


    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


    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


    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


    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


    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


    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


    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


    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


    Answers and Hints for Solution of Exercises


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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