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. Cover......Page 1 Complex Variables and Applications, Seventh Edition......Page 2 Preface......Page 7 Sums and Products......Page 11 Basic Algebraic Properties......Page 13 Further Properties......Page 15 Moduli......Page 18 Complex Conjugates......Page 21 Exponential Form......Page 25 Products and Quotients in Exponential Form......Page 27 Roots of Complex Numbers......Page 32 Examples......Page 35 Regions in the Complex Plane......Page 39 Functions of a Complex Variable......Page 43 Mappings......Page 46 Mappings by the Exponential Function......Page 50 Limits......Page 53 Theorems on Limits......Page 56 Limits Involving the Point at Infinity......Page 58 Continuity......Page 61 Derivatives......Page 64 Differentiation Formu]as......Page 67 Cauchy-Riemann Equations......Page 70 Sufficient Conditions for Differentiability......Page 73 Polar Coordinates......Page 75 Analytic Functions......Page 80 Examples......Page 82 Harmonic Functions......Page 85 Uniquely Determined Analytic Functions......Page 90 Reflection Principle......Page 92 The Exponential Function......Page 97 The Logarithmic Function......Page 100 Branches and Derivatives of Logarithms......Page 102 Some Identities Involving Logarithms......Page 105 Complex Exponents......Page 107 Trigonometric Functions......Page 110 Hyperbolic Functions......Page 115 Inverse Trigonometric and Hyperbolic Functions......Page 118 Derivatives of Functions w(t)......Page 121 Definite Integrals of Functions w(t)......Page 123 Contours......Page 126 Contour Integrals......Page 132 Examples......Page 134 Upper Bounds for Moduli of Contour Integrals......Page 140 Antiderivatives......Page 145 Examples......Page 148 Cauchy-Goursat Theorem......Page 152 Proof of the Theorem......Page 154 Simply and Multiply Connected Domains......Page 159 Cauchy Integral Formula......Page 167 Derivatives of Analytic Functions......Page 168 Liouville's Theorem and the Fundamental Theorem of Algebra......Page 175 Maximum Modulus Principle......Page 177 Convergence of Sequences......Page 185 Convergence of Series......Page 188 Taylor Series......Page 192 Examples......Page 195 Laurent Series......Page 200 Examples......Page 205 Abso1ute and Uniform Convergence of Power Series......Page 210 Continuity of Sums of Power Series......Page 214 Integration and Differentiation of Power Series......Page 216 Uniqueness of Series Representations......Page 220 Multiplication and Division of Power Series......Page 225 Residues......Page 231 Cauchy's Residue Theorem......Page 235 Using a Single Residue......Page 237 The Three TYpes of Isolated Singular Points......Page 241 Residues at Poles......Page 244 Examples......Page 246 Zeros of Analytic Functions......Page 249 Zeros and Poles......Page 252 Behavior off Near Isolated Singular Points......Page 257 Evaluation of Improper Integrals......Page 261 Example......Page 264 Improper Integrals from Fourier Analysis......Page 269 Jordan's Lemma......Page 272 Indented Paths......Page 277 An Indentation Around a Branch Point......Page 280 Integration Along a Branch Cut......Page 283 Definite Integrals involving Sines and Cosines......Page 288 Argument Principle......Page 291 Rouche's Theorem......Page 294 Inverse Laplace Transforms......Page 298 Examples......Page 301 Linear Transformations......Page 309 The Transformation w = l/z......Page 311 Mappings by 1/z......Page 313 Linear Fractional Transformations......Page 317 An Implicit Form......Page 320 Mappings of the Upper Half Plane......Page 323 The Transformation w = sin z......Page 328 Mappings by z^2 and Branches of z^1/2......Page 334 Square Roots of Polynomials......Page 339 Riemann Surfaces......Page 345 Surfaces for Related Functions......Page 348 Preservation of Angles......Page 353 Scale Factors......Page 356 Local Inverses......Page 358 Harmonic Conjugates......Page 361 Transformations of Harmonic Functions......Page 363 Transformations of Boundary Conditions......Page 365 Steady Temperatures......Page 371 Steady Temperatures in a Half Plane......Page 373 A Related Problem......Page 375 Temperatures in a Quadrant......Page 378 Electrostatic Potential......Page 383 Potential in a Cylindrical Space......Page 384 Two-Dimensional Fluid Flow......Page 389 The Stream Function......Page 391 Flows Around a Corner and Around a Cylinder......Page 393 Mapping the Real Axis onto a Polygon......Page 401 Schwarz-Christoffel Transformation......Page 403 Triangles and Rectangles......Page 407 Degenerate Polygons......Page 411 Fluid Flow in a Channel Through a Slit......Page 416 Flow in a Channel with an Offset......Page 418 Electrostatic Potential about an Edge of a Conducting Plate......Page 421 Poisson Integral Formula......Page 427 Dirichlet Problem for a Disk......Page 429 Related Boundary Value Problems......Page 433 Schwarz Integral Formula......Page 437 Dirichlet Problem for a Half Plane......Page 439 Neumann Problems......Page 443 Bibliography......Page 447 Table of Transformations of Regions......Page 451 Index......Page 461 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