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.
Author(s): James Brown, Ruel Churchill
Series: Brown and Churchill
Edition: 9
Publisher: McGraw-Hill Education
Year: 2013
Language: English
Pages: C, xvi, 461
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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
