复变函数及应用 第9版 英文版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 Complex Numbers1

Sums and Products1

Basic Algebraic Properties3

Further Algebraic Properties5

Vectors and Moduli8

Triangle Inequality11

Complex Conjugates14

Exponential Form17

Products and Powers in Exponential Form20

Arguments of Products and Quotients21

Roots of Complex Numbers25

Examples28

Regions in the Complex Plane32

2 Analytic Functions37

Functions and Mappings37

The Mapping w＝z240

Limits44

Theorems on Limits47

Limits Involving the Point at Infinity50

Continuity52

Derivatives55

Rules for Differentiation59

Cauchy-Riemann Equations62

Examples64

Sufficient Conditions for Differentiability65

Polar Coordinates68

Analytic Functions72

Further Examples74

Harmonic Functions77

Uniquely Determined Analytic Functions80

Reflection Principle82

3 Elementary Functions87

The Exponential Function87

The Logarithmic Function90

Examples92

Branches and Derivatives of Logarithms93

Some Identities Involving Logarithms97

The Power Function100

Examples101

The Trigonometric Functions sin z and cos z103

Zeros and Singularities of Trigonometric Functions105

Hyperbolic Functions109

Inverse Trigonometric and Hyperbolic Functions112

4 Integrals115

Derivatives of Functions w(t)115

Definite Integrals of Functions w(t)117

Contours120

Contour Integrals125

Some Examples127

Examples Involving Branch Cuts131

Upper Bounds for Moduli of Contour Integrals135

Antiderivatives140

Proof of the Theorem144

Cauchy-Goursat Theorem148

Proof of the Theorem150

Simply Connected Domains154

Multiply Connected Domains156

Cauchy Integral Formula162

An Extension of the Cauchy Integral Formula164

Verification of the Extension166

Some Consequences of the Extension168

Liouville's Theorem and the Fundamental Theorem of Algebra172

Maximum Modulus Principle173

5 Series179

Convergence of Sequences179

Convergence of Series182

Taylor Series186

Proof of Taylor's Theorem187

Examples189

Negative Powers of(z-z0)193

Laurent Series197

Proof of Laurent's Theorem199

Examples202

Absolute and Uniform Convergence of Power Series208

Continuity of Sums of Power Series211

Integration and Differentiation of Power Series213

Uniqueness of Series Representations216

Multiplication and Division of Power Series221

6 Residues and Poles227

Isolated Singular Points227

Residues229

Cauchy's Residue Theorem233

Residue at Infinity235

The Three Types of Isolated Singular Points238

Examples240

Residues at Poles242

Examples244

Zeros of Analytic Functions248

Zeros and Poles251

Behavior of Functions Near Isolated Singular Points255

7 Applications of Residues259

Evaluation of Improper Integrals259

Example262

Improper Integrals from Fourier Analysis267

Jordan's Lemma269

An Indented Path274

An Indentation Around a Branch Point277

Integration Along a Branch Cut280

Definite Integrals Involving Sines and Cosines284

Argument Principle287

Rouché's Theorem290

Inverse Laplace Transforms294

8 Mapping by Elementary Functions299

Linear Transformations299

The Transformation w＝1/z301

Mappings by 1/z303

Linear Fractional Transformations307

An Implicit Form310

Mappings of the Upper Half Plane313

Examples315

Mappings by the Exponential Function318

Mapping Vertical Line Segments by w＝sin z320

Mapping Horizontal Line Segments by w＝sin z322

Some Related Mappings324

Mappings by z2326

Mappings by Branches of z1/2328

Square Roots of Polynomials332

Riemann Surfaces338

Surfaces for Related Functions341

9 Conformal Mapping345

Preservation of Angles and Scale Factors345

Further Examples348

Local Inverses350

Harmonic Conjugates354

Transformations of Harmonic Functions357

Transformations of Boundary Conditions360

10 Applications of Conformal Mapping365

Steady Temperatures365

Steady Temperatures in a Half Plane367

A Related Problem369

Temperatures in a Quadrant371

Electrostatic Potential376

Examples377

Two-Dimensional Fluid Flow382

The Stream Function384

Flows Around a Corner and Around a Cylinder386

11 The Schwarz-Christoffel Transformation393

Mapping the Real Axis onto a Polygon393

Schwarz-Christoffel Transformation395

Triangles and Rectangles399

Degenerate Polygons402

Fluid Flow in a Channel through a Slit407

Flow in a Channel with an Offset409

Electrostatic Potential about an Edge of a Conducting Plate412

12 Integral Formulas of the Poisson Type417

Poisson Integral Formula417

Dirichlet Problem for a Disk420

Examples422

Related Boundary Value Problems426

Schwarz Integral Formula428

Dirichlet Problem for a Half Plane430

Neumann Problems433

Appendixes437

Bibliography437

Table of Transformations of Regions441

Index451