Complex Variables and Applications / Edition 8

Complex Variables and Applications / Edition 8

by James Brown
Pub. Date:
McGraw-Hill Higher Education


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Complex Variables and Applications / Edition 8

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.

Product Details

ISBN-13: 2900073051948
Publisher: McGraw-Hill Higher Education
Publication date: 01/10/2008
Series: Brown and Churchill Series
Edition description: Older Edition
Pages: 480
Product dimensions: 6.50(w) x 1.50(h) x 9.50(d)

About the Author

RUEL V. CHURCHILL (University of Michigan)

Table of Contents

Preface xv

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

Rouché'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

Riemarm 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

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