复变函数及应用 第7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 Properties5

Moduli8

Complex Conjugates11

Exponential Form15

Products and Quotients in Exponential Form17

Roots of Complex Numbers22

Examples25

Regions in the Complex Plane29

2 Analytic Functions33

Functions of a Complex Variable33

Mappings36

Mappings by the Exponential Function40

Limits43

Theorems on Limits46

Limits Involving the Point at Infinity48

Continuity51

Derivatives54

Differentiation Formulas57

Cauchy-Riemann Equations60

Sufficient Conditions for Differentiability63

Polar Coordinates65

Analytic Functions70

Examples72

Harmonic Functions75

Uniquely Determined Analytic Functions80

Reflection Principle82

3 Elementary Functions87

The Exponential Function87

The Logarithmic Function90

Branches and Derivatives of Logarithms92

Some Identities Involving Logarithms95

Complex Exponents97

Trigonometric Functions100

Hyperbolic Functions105

Inverse Trigonometric and Hyperbolic Functions108

4 Integrals111

Derivatives of Functions w（t）111

Definite Integrals of Functions w（t）113

Contours116

Contour Integrals122

Examples124

Upper Bounds for Moduli of Contour Integrals130

Antiderivatives135

Examples138

Cauchy-Goursat Theorem142

Proof of the Theorem144

Simply and Multiply Connected Domains149

Cauchy Integral Formula157

Derivatives of Analytic Functions158

Liouville’s Theorem and the Fundamental Theorem of Algebra165

Maximum Modulus Principle167

5 Series175

Convergence of Sequences175

Convergence of Series178

Taylor Series182

Examples185

Laurent Series190

Examples195

Absolute and Uniform Convergence of Power Series200

Continuity of Sums of Power Series204

Integration and Differentiation of Power Series206

Uniqueness of Series Representations210

Multiplication and Division of Power Series215

6 Residues and Poles221

Residues221

Cauchy’s Residue Theorem225

Using a Single Residue227

The Three Types of Isolated Singular Points231

Residues at Poles234

Examples236

Zeros of Analytic Functions239

Zeros and Poles242

Behavior off Near Isolated Singular Points247

7 Applications of Residues251

Evaluation of Improper Integrals251

Example254

Improper Integrals from Fourier Analysis259

Jordan’s Lemma262

Indented Paths267

An Indentation Around a Branch Point270

Integration Along a Branch Cut273

Definite Integrals involving Sines and Cosines278

Argument Principle281

Rouche’s Theorem284

Inverse Laplace Transforms288

Examples291

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

The Transformation w = sin z318

Mappings by z2 and Branches of z 1/2324

Square Roots of Polynomials329

Riemann Surfaces335

Surfaces for Related Functions338

9 Conformal Mapping343

Preservation of Angles343

Scale Factors346

Local Inverses348

Harmonic Conjugates351

Transformations of Harmonic Functions353

Transformations of Boundary Conditions355

10 Applications of Conformal Mapping361

Steady Temperatures361

Steady Temperatures in a Half Plane363

A Related Problem365

Temperatures in a Quadrant368

Electrostatic Potential373

Potential in a Cylindrical Space374

Two-Dimensional Fluid Flow379

The Stream Function381

Flows Around a Corner and Around a Cylinder383

11 The Schwarz—Christoffel Transformation391

Mapping the Real Axis onto a Polygon391

Schwarz-Christoffel Transformation393

Triangles and Rectangles397

Degenerate Polygons401

Fluid Flow in a Channel Through a Slit406

Flow in a Channel with an Offset408

Electrostatic Potential about an Edge of a Conducting Plate411

12 Integral Formulas of the Poisson Type417

Poisson Integral Formula417

Dirichlet Problem for a Disk419

Related Boundary Value Problems423

Schwarz Integral Formula427

Dirichlet Problem for a Half Plane429

Neumann Problems433

Appendixes437

Bibliography437

Table of Transformations of Regions441

Index451