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

1 Complex Numbers 1 Sums and Products 1 Basic Algebraic Properties 3 Further Algebraic Properties 5 Vectors and Moduli 8 Triangle Inequality 11 Complex Conjugates 14 Exponential Form 17 Products and Powers in Exponential Form 20 Arguments of Products and Quotients 21 Roots of Complex Numbers 25 Examples 28 Regions in the Complex Plane 32 2 Analytic Functions 37 Functions and Mappings 37 The Mapping w = z2 40 Limits 44 Theorems on Limits 47 Limits Involving the Point at Infinity 50 Continuity 52 Derivatives 55 Rules for Differentiation 59 Cauchy-Riemann Equations 62 Examples 64 Sufficient Conditions for Differentiability 65 Polar Coordinates 68 Analytic Functions 72 Further Examples 74 Harmonic Functions 77 Uniquely Determined Analytic Functions 80 Reflection Principle 82 3 Elementary Functions 87 The Exponential Function 87 The Logarithmic Function 90 Examples 92 Branches and Derivatives of Logarithms 93 Some Identities Involving Logarithms 97 The Power Function 100 Examples 101 The Trigonometric Functions sin z and cos z 103 Zeros and Singularities of Trigonometric Functions 105 Hyperbolic Functions 109 Inverse Trigonometric and Hyperbolic Functions 112 4 Integrals 115 Derivatives of Functions w(t) 115 Definite Integrals of Functions w(t) 117 Contours 120 Contour Integrals 125 Some Examples 127 Examples Involving Branch Cuts 131 Upper Bounds for Moduli of Contour Integrals 135 Antiderivatives 140 Proof of the Theorem 144 Cauchy-Goursat Theorem 148 Proof of the Theorem 150 Simply Connected Domains 154 Multiply Connected Domains 156 Cauchy Integral Formula 162 An Extension of the Cauchy Integral Formula 164 Verification of the Extension 166 Some Consequences of the Extension 168 Liouville's Theorem and the Fundamental Theorem of Algebra 172 Maximum Modulus Principle 173 5 Series 179 Convergence of Sequences 179 Convergence of Series 182 Taylor Series 186 Proof of Taylor's Theorem 187 Examples 189 Negative Powers of (z − z0) 193 Laurent Series 197 Proof of Laurent's Theorem 199 Examples 202 Absolute and Uniform Convergence of Power Series 208 Continuity of Sums of Power Series 211 Integration and Differentiation of Power Series 213 Uniqueness of Series Representations 216 Multiplication and Division of Power Series 221 6 Residues and Poles 227 Isolated Singular Points 227 Residues 229 Cauchy's Residue Theorem 233 Residue at Infinity 235 The Three Types of Isolated Singular Points 238 Examples 240 Residues at Poles 242 Examples 244 Zeros of Analytic Functions 248 Zeros and Poles 251 Behavior of Functions Near Isolated Singular Points 255 7 Applications of Residues 259 Evaluation of Improper Integrals 259 Example 262 Improper Integrals from Fourier Analysis 267 Jordan's Lemma 269 An Indented Path 274 An Indentation Around a Branch Point 277 Integration Along a Branch Cut 280 Definite Integrals Involving Sines and Cosines 284 Argument Principle 287 Rouche's Theorem 290 Inverse Laplace Transforms 294 8 Mapping by Elementary Functions 299 Linear Transformations 299 The Transformation w = 1/z 301 Mappings by 1/z 303 Linear Fractional Transformations 307 An Implicit Form 310 Mappings of the Upper Half Plane 313 Examples 315 Mappings by the Exponential Function 318 Mapping Vertical Line Segments by w = sin z 320 Mapping Horizontal Line Segments by w = sin z 322 Some Related Mappings 324 Mappings by z2 326 Mappings by Branches of z1/2 328 Square Roots of Polynomials 332 Riemann Surfaces 338 Surfaces for Related Functions 341 9 Conformal Mapping 345 Preservation of Angles and Scale Factors 345 Further Examples 348 Local Inverses 350 Harmonic Conjugates 354 Transformations of Harmonic Functions 357 Transformations of Boundary Conditions 360 10 Applications of Conformal Mapping 365 Steady Temperatures 365 Steady Temperatures in a Half Plane 367 A Related Problem 369 Temperatures in a Quadrant 371 Electrostatic Potential 376 Examples 377 Two-Dimensional Fluid Flow 382 The Stream Function 384 Flows Around a Corner and Around a Cylinder 386 11 The Schwarz-Christoffel Transformation 393 Mapping the Real Axis onto a Polygon 393 Schwarz-Christoffel Transformation 395 Triangles and Rectangles 399 Degenerate Polygons 402 Fluid Flow in a Channel through a Slit 407 Flow in a Channel with an Offset 409 Electrostatic Potential about an Edge of a Conducting Plate 412 12 Integral Formulas of the Poisson Type 417 Poisson Integral Formula 417 Dirichlet Problem for a Disk 420 Examples 422 Related Boundary Value Problems 426 Schwarz Integral Formula 428 Dirichlet Problem for a Half Plane 430 Neumann Problems 433 Appendixes 437 Bibliography 437 Table of Transformations of Regions 441 Index 451

James Ward Brown & Ruel V. Churchill 

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