Complex Variables and Applications

"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. 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 Preface -- 1. Complex Numbers -- 2. Analytic Functions -- 3. Elementary Functions -- 4. Integrals -- 5. Series -- 6. Residues And Poles -- 7. Applications Of Residues -- 8. Mapping By Elementary Functions -- 9. Conformal Mapping -- 10. Applications Of Conformal Mapping -- 11. The Schwarz-christoffel Transformation -- 12. Integral Formulas Of The Poisson Type -- Appendixes -- Index. James Ward Brown, Ruel V. Churchill. Churchill Listed As Main Author In Previous Edition. Includes Bibliographical References (p. 437-439) And Index. From the publisher: Complex Variables and Applications provides a one-term introduction to the theory and application of functions of a complex variable. Its primary objective is to develop those parts of the theory that are prominent in the applications of the subject. Numerous applications to the physical sciences and engineering are provided throughout, including those suitable for reference and self-study