"Complex Variables and Applications, 8E" 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 Ward Brown, Ruel V. Churchill
Edition: 7th
Publisher: McGraw Hill Higher Education
Year: 2003
Language: English
Commentary: pages(i, ii, iv, v, vi, vii, viii) are missing
Pages: xviii+458
Preface
Chapter 1 Complex Numbers
1 Sums and Products
2 Basic Algebraic Properties
3 Further Properties
4 Moduli
5 Complex Conjugates
6 Exponential Fonn
7 Products and Quotients in Exponential Form
8 Roots of Complex Numbers
9 Examples
10 Regions in the Complex Plane
Chapter 2 Analytic Functions
11 Functions of a Complex Variable
12 Mappings
13 Mappings by the Exponential Function
14 Limits
15 Theorems on Limits
16 Limits Involving the Point at Infinity
17 Continuity
18 Derivatives
19 Differentiation Formulas
20 Cauchy-Riemann Equations
21 Sufficient Conditions for Differentiability
22 Polar Coordinates
23 Analytic Functions
24 Examples
25 Harmonic Functions
26 Uniquely Determined Analytic Functions
27 Reflection Principle
Chapter 3 Elementary Functions
28 The Exponential Function
29 The Logarithmic Function
30 Branches and Derivatives of Logarithms
31 Some Identities Involving Logarithms
32 Complex Exponents
33 Trigonometric Functions
34 Hyperbolic Functions
35 Inverse Trigonometric and Hyperbolic Functions
Chapter 4 Integrals
36 Derivatives of Functions w(t)
37 Definite Integrals of Functions w(t)
38 Contours
39 Contour Integrals
40 Examples
41 Upper Bounds for Moduli of Contour Integrals
42 Antiderivatives
43 Examples
44 Cauchy-Goursat Theorem
45 Proof of the Theorem
46 Simply and Multiply Connected Domains
47 Cauchy Integral Formula
48 Derivatives of Analytic Functions
49 Liouville's Theorem and the Fundamental Theorem of Algebra
50 Maximum Modulus Principle
Chapter 5 Series
51 Convergence of Sequences
52 Convergence of Series
53 Taylor Series
54 Examples
55 Laurent Series
56 Examples
57 Absolute and Uniform Convergence of Power Series
58 Continuity of Sums of Power Series
59 Integration and Differentiation of Power Series
60 Uniqueness of Series Representations
61 Multiplication and Division of Power Series
Chapter 6 Residues and Poles
62 Residues
63 Cauchy's Residue Theorem
64 Using a Single Residue
65 The Three Types of Isolated Singular Points
66 Residues at Poles
67 Examples
68 Zeros of Analytic Functions
69 Zeros and Poles
70 Behavior off Near Isolated Singular Points
Chapter 7 Applications of Residues
71 Evaluation of Improper Integrals
72 Example
73 Improper Integrals from Fourier Analysis
74 Jordan's Lemma
75 Indented Paths
76 An Indentation Around a Branch Point
77 Integration Along a Branch Cut
78 Definite Integrals involving Sines and Cosines
79 Argument Principle
80 Roucht's Theorem
81 Inverse Laplace Transforms
82 Examples
Chapter 8 Mapping by Elementary Functions
83 Linear Transformations
84 The Transformation w = l/z
85 Mappings by 1/z
86 Linear Fractional Transformations
87 An Implicit Fonn
88 Mappings of the Upper Half Plane
89 The Transformation w = sin z
90 Mappings by z^2 and Branches of z^{1/2}
91 Square Roots of Polynomials
92 Riemann Surfaces
93 Surfaces for Related Functions
Chapter 9 Conformal Mapping
94 Preservation of Angles
95 Scale Factors
96 Local Inverses
97 Harmonic Conjugates
98 Transformations of Harmonic Functions
99 Transformations of Boundary Conditions
Chapter 10 Applications of Conformal Mapping
100 Steady Temperatures
101 Steady Temperatures in a Half Plane
102 A Related Problem
103 Temperatures in a Quadrant
104 Electrostatic Potential
105 Potential in a Cylindrical Space
106 Two-Dimensional Fluid How
107 The Stream Function
108 Flows Around a Comer and Around a Cylinder
Chapter 11 The Schwarz-Christoffel Transformation
109 Mapping the Real Axis onto a Polygon
110 Schwarz-Christoffel Transformation
111 Triangles and Rectangles
112 Degenerate Polygons
113 Fluid Flow in a Channel Through a Slit
114 Flow in a Channel with an Offset
115 Electrostatic Potential about an Edge of a Conducting Plate
Chapter 12 Integral Formulas of the Poisson Type
116 Poisson Integral Formula
117 Dirichlet Problem for a Disk
118 Related Boundary Value Problems
119 Schwarz Integral Formula
120 Dirichlet Problem for a Half Plane
121 Neumann Problems
Appendix 1 Bibliography
Appendix 2 Table of Transformations of Regions
Index
