Complex Variables and Applications

Description

Complex Variables and Applications, 9e will serve, just as the earlier editions did, as a textbook for an introductory course in the theory and application of functions of a complex variable. This new edition preserves the basic content and style of the earlier editions. The text is designed to develop the theory that is prominent in applications of the subject. You will find a special emphasis given to the application of residues and conformal mappings. To accommodate the different calculus backgrounds of students, footnotes are given with references to other texts that contain proofs and discussions of the more delicate results in advanced calculus. Improvements in the text include extended explanations of theorems, greater detail in arguments, and the separation of topics into their own sections. PREFACE CHAPTER 1 COMPLEX NUMBERS 1. SUMS AND PRODUCTS 2. BASIC ALGEBRAIC PROPERTIES 3. FURTHER ALGEBRAIC PROPERTIES 4. VECTORS AND MODULI 5. TRIANGLE INEQUALITY 6. COl\ilPLEX CONJUGATES 7. EXPONENTIAL FORM 8. PRODUCTS AND POWERS IN EXPONENTIAL FORM 9. ARGUMENTS OF PRODUCTS AND QUOTIENTS 10. ROOTS OF COMPLEX NUMBERS 11. EXAMPLES 12. REGIONS IN THE COMPLEX PLANE CHAPTER 2 ANALYTIC FUNCTIONS 13. FUNCTIONS AND MAPPINGS 14. THE MAPPING w = z^2 15. LlMITS 16. THEOREMS ON LIMITS 17. LIMITS INVOLVING THE POINT AT INFINITY 18. CONTINUITY 19. DERIVATIVES 20. RULES FOR DIFFERENTIATION 21. CAUCHY-RIEMANN EQUATIONS 22. EXAMPLES 23. SUFFICIENT CONDITIONS FOR DIFFERENTIABILITY 24. POLAR COORDINATES 25. ANALYTIC FUNCTIONS 26. FURTHER EXAMPLES 27. HARMONIC FUNCTIONS 28. UNIQUELY DETERMINED ANALYTIC FUNCTIONS 29. REFLECTION PRINCIPLE CHAPTER 3 ELEMENTARY FUNCTIONS 30. THE EXPONENTIAL FUNCTION 31. THE LOGARITHMIC FUNCTION 32. EXAMPLES 33. BRANCHES AND DERIVATIVES OF LOGARITHMS 34. SOME IDENTITIES INVOLVING LOGARITHIMS 35. THE POWER FUNCTION 36. EXAMPLES 37. THE TRIGONOMETRIC FUNCTIONS sin z AND cos z 38. ZEROS AND SINGULARITIES OF TRIGONOMETRIC FUNCTIONS 39. HYPERBOLIC FUNCTIONS 40. INVERSE TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS CHAPTER 4 INTEGRALS 41. DERIVATIVES OF FUNCTIONS w(t) 42. DEFINITE INTEGRALS OF FUNCTIONS w(t) 43. CONTOURS 44. CONTOUR INTEGRALS 45. SOME EXAMPLES 46. EXAMPLES INVOLVING BRANCH CUTS 47. UPPER BOUNDS FOR MODULI OF CONTOUR INTEGRALS 48. ANTIDERIVATIVES 49. PROOF OF THE THEOREM 50. CAUCHY-GOURSAT THEOREM 51. PROOF OF THE THEOREM 52. SlMPLY CONNECTED DOMAINS 53. MULTIPLY CONNECTED DOMAINS 54. CAUCHY INTEGRAL FORMULA 55. AN EXTENSION OF THE CAUCHY INTEGRAL FORMULA 56. VERIFICATION OF THE EXTENSION 57. SOME CONSEQUENCES OF THE EXTENSION 58. LIOUVILLE'S THEOREM AND THE FUNDAIVIENTALTHEOREM OF ALGEBRA 59. MAXIMUM MODULUS PRINCIPLE CHAPTER 5 SERIES 60. CONVERGENCE OF SEQUENCES 61. CONVERGENCE OF SERIES 62. TAYLOR SERIES 63. PROOF OF TAYLOR'S THEOREM 64. EXAMPLES 65. NEGATIVE POWERS OF (z - zo) 66. LAURENT SERIES 67. PROOF OF LAURENT'S THEOREM 68. EXAMPLES 69. ABSOLUTE AND UNIFORIVI CONVERGENCE OF POWER SERIES 70. CONTINUITY OF SUMS OF POWER SERIES 71. INTEGRATION AND DIFFERENTIATION OF POWER SERIES 72. UNIQUENESS OF SERIES REPRESENTATIONS 73. MULTIPLICATION AND DIVISION OF POWER SERIES CHAPTER 6 RESIDUES AND POLES 74. ISOLATED SINGULAR POINTS 75. RESIDUES 76. CAUCHY'S RESIDUE THEOREM 77. RESIDUE AT INFINITY 78. THE THREE TYPES OF ISOLATED SINGULAR POINTS 79. EXAMPLES 80. RESIDUES AT POLES 81. EXAMPLES 82. ZEROS OF ANALYTIC FUNCTIONS 83. ZEROS AND POLES 84. BEHAVIOR OF FUNCTIONS NEAR ISOLATED SINGULAR POINTS CHAPTER 7 APPLICATIONS OF RESIDUES 85. EVALUATION OF IMPROPER INTEGRALS 86. EXAMPLE 87. IMPROPER INTEGRALS FROM FOURIER ANALYSIS 88. JORDAN'S LEMMA 89. AN INDENTED PATH 90. AN INDENTATION AROUND A BRANCH POINT 91. INTEGRATION ALONG A BRANCH CUT 92. DEFINITE INTEGRALS INVOLVING SINES AND COSINES 93. ARGUMENT PRINCIPLE 94. ROUCHE'S THEOREM 95. INVERSE LAPLACE TRANSFORMS CHAPTER 8 MAPPING BY ELEMENTARY FUNCTIONS 96. LINEAR TRANSFORMATIONS 97. THE TRANSFORMATION w = 1/z 98. MAPPINGS BY 1/z 99. LINEAR FRACTIONAL TRANSFORMATIONS 100. AN IMPLICIT FORM 101. MAPPINGS OF THE UPPER HALF PLANE 102. EXAMPLES 103. MAPPINGS BY THE EXPONENTIAL FUNCTION 104. MAPPING VERTICAL LINE SEGMENTS BY w = sinz 105. MAPPING HORIZONTAL LINESEGMENTS BY w =sin z 106. SOME RELATED MAPPINGS 107. MAPPINGS BY z^2 108. MAPPINGS BY BRANCHES OF z^(1/2) 109. SQUARE ROOTS OF POLYNOMIALS 110. RIEMANN SURFACES 111. SURFACES FOR RELATED FUNCTIONS CHAPTER 9 CONFORMAL MAPPING 112. PRESERVATION OF ANGLES AND SCALE FACTORS 113. FURTHER EXAMPLES 114. LOCAL INVERSES 115. HARMONIC CONJUGATES 116. TRANSFORMATIONS OF HARMONIC FUNCTIONS 117. TRANSFORMATIONS OF BOUNDARY CONDITIONS CHAPTER 10 APPLICATIONS OF CONFORMAL MAPPING 118. STEADY TEMPERATURES 119. STEADY TEMPERATURES IN A HALF PLANE 120. A RELATED PROBLEM 121. TElMPERATURES IN A QUADRANT 122. ELECTROSTATIC POTENTIAL 123. EXAMPLES 124. TWO-DlMENSIONAL FLUID FLOW 125. THE STREAM FUNCTION 126. FLOWS AROUND A CORNER AND AROUND A CYLINDER CHAPTER 11 THE SCHWARZ-CHRISTOFFEL TRANSFORMATION 127. MAPPING THE REAL AXIS ONTO A POLYGON 128. SCHWARZ-CHRISTOFFEL TRANSFORMATION 129. TRIANGLES AND RECTANGLES 130. DEGENERATE POLYGONS 131. FLUID FLOW IN A CHANNEL THROUGH A SLIT 132. FLOW IN A CHANNEL WITH AN OFFSET 133. ELECTROSTATIC POTENTIAL ABOUT AN EDGE OF A CONDUCTING PLATE CHAPTER 12 INTEGRAL FORMULAS OF THE POISSON TYPE 134. POISSON INTEGRAL FORMULA 135. DIRICHLET PROBLEM FOR A DISK 136. EXA.MPLES 137. RELATED BOUNDARY VALUE PROBLEMS 138. SCHWARZ INTEGRAL FORMULA 139. DIRICHLET PROBLEM FOR A HALF PLANE 140. NEUMANN PROBLEMS APPENDIX 1 BIBLIOGRAPHY APPENDIX 2 TABLE OF TRANSFORMATIONS OF REGIONS (See Chap. 8) INDEX