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 Title page......Page 3 CONTENTS......Page 5 Preface......Page 9 Sums and Products......Page 13 Basic Algebraic Properties......Page 15 Further Properties......Page 17 Moduli......Page 20 Complex Conjugates......Page 23 Exponential Form......Page 27 Products and Quotients in Exponential Form......Page 29 Roots of Complex Numbers......Page 34 Examples......Page 37 Regions in the Complex Plane......Page 41 Functions of a Complex Variable......Page 45 Mappings......Page 48 Mappings by the Exponential Function......Page 52 Limits......Page 55 Theorems on Limits......Page 58 Limits Involving the Point at Infinity......Page 60 Continuity......Page 63 Derivatives......Page 66 Differentiation Formulas......Page 69 Cauchy-Riemann Equations......Page 72 Sufficient Conditions for Differentiability......Page 75 Polar Coordinates......Page 77 Analytic Functions......Page 82 Examples......Page 84 Harmonic Functions......Page 87 Uniquely Determined Analytic Functions......Page 92 Reflection Principle......Page 94 The Exponential Function......Page 99 The Logarithmic Function......Page 102 Branches and Derivatives of Logarithms......Page 104 Some Identities Involving Logarithms......Page 107 Complex Exponents......Page 109 Trigonometric Functions......Page 112 Hyperbolic Functions......Page 117 Inverse Trigonometric and Hyperbolic Functions......Page 120 Derivatives of Functions $w(t)$......Page 123 Definite Integrals of Functions $w(t)$......Page 125 Contours......Page 128 Contour Integrals......Page 134 Examples......Page 136 Upper Bounds for Moduli of Contour Integrals......Page 142 Antiderivatives......Page 147 Examples......Page 150 Cauchy-Goursat Theorem......Page 154 Proof of the Theorem......Page 156 Simply and Multiply Connected Domains......Page 161 Cauchy Integral Formula......Page 169 Derivatives of Analytic Functions......Page 170 Liouville's Theorem and the Fundamental Theorem of Algebra......Page 177 Maximum Modulus Principle......Page 179 Convergence of Sequences......Page 187 Convergence of Series......Page 190 Taylor Series......Page 194 Examples......Page 197 Laurent Series......Page 202 Examples......Page 207 Absolute and Uniform Convergence of Power Series......Page 212 Continuity of Sums of Power Series......Page 216 Integration and Differentiation of Power Series......Page 218 Uniqueness of Series Representations......Page 222 Multiplication and Division of Power Series......Page 227 Residues......Page 233 Cauchy's Residue Theorem......Page 237 Using a Single Residue......Page 239 The Three Types of Isolated Singular Points......Page 243 Residues at Poles......Page 246 Examples......Page 248 Zeros of Analytic Functions......Page 251 Zeros and Poles......Page 254 Behavior of $f$ Near Isolated Singular Points......Page 259 Evaluation of Improper Integrals......Page 263 Example......Page 266 Improper Integrals from Fourier Analysis......Page 271 Jordan's Lemma......Page 274 Indented Paths......Page 279 An Indentation Around a Branch Point......Page 282 Integration Along a Branch Cut......Page 285 Definite Integrals involving Sines and Cosines......Page 290 Argument Principle......Page 293 Rouche's Theorem......Page 296 Inverse Laplace Transforms......Page 300 Examples......Page 303 Linear Transformations......Page 311 The Transformation $w = 1/z$......Page 313 Mappings by $1/z$......Page 315 Linear Fractional Transformations......Page 319 An Implicit Form......Page 322 Mappings of the Upper Half Plane......Page 325 The Transformation $w = \sin z$......Page 330 Mappings by $z^2$ and Branches of $z^{1/2}$......Page 336 Square Roots of Polynomials......Page 341 Riemann Surfaces......Page 347 Surfaces for Related Functions......Page 350 Preservation of Angles......Page 355 Scale Factors......Page 358 Local Inverses......Page 360 Harmonic Conjugates......Page 363 Transformations of Harmonic Functions......Page 365 Transformations of Boundary Conditions......Page 367 Steady Temperatures......Page 373 Steady Temperatures in a Half Plane......Page 375 A Related Problem......Page 377 Temperatures in a Quadrant......Page 380 Electrostatic Potential......Page 385 Potential in a Cylindrical Space......Page 386 Two-Dimensional Fluid Flow......Page 391 The Stream Function......Page 393 Flows Around a Corner and Around a Cylinder......Page 395 Mapping the Real Axis onto a Polygon......Page 403 Schwarz-Christoffel Transformation......Page 405 Triangles and Rectangles......Page 409 Degenerate Polygons......Page 413 Fluid Flow in a Channel Through a Slit......Page 418 Flow in a Channel with an Offset......Page 420 Electrostatic Potential about an Edge of a Conducting Plate......Page 423 Poisson Integral Formula......Page 429 Dirichlet Problem for a Disk......Page 431 Related Boundary Value Problems......Page 435 Schwarz Integral Formula......Page 439 Dirichlet Problem for a Half Plane......Page 441 Neumann Problems......Page 445 Bibliography......Page 449 Table of Transformations of Regions......Page 453 Index......Page 463 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