Functions of a Complex Variable and Some of Their Applications

From the Foreword to the First EditionForeword to the English EditionIntroduction 1. Complex Numbers 2. The Simplest Operations 3. Multiplication, Division, Integral Powers and Roots 4. Complex Powers. Logarithms ExercisesI. 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 ExercisesII. 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 ExercisesIII. Elementary Functions 24. The Functions w = zn and their Riemann Surfaces 25. The Concept of a Regular Branch. The Functions w = n√z 26. The Function w = 1/2(z+z-1) 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 ExercisesIV. 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 ExercisesV. 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 ExercisesVI. 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 ExercisesVII. 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}αx. dx 74. Other Integrals 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 ExercisesVIII. Mapping of Polygonal Domains 82. The Symmetry Principle 83. Illustrative Examples 84. The Schwarz-Christoffel 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 ExercisesAnswers and Hints for Solution of ExercisesIndex