复函数论导论 英文2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

复函数论导论 英文PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Ⅰ The Complex Number System1

1 The Algebra and Geometry of Complex Numbers1

1.1 The Field of Complex Numbers1

1.2 Conjugate,Modulus,and Argument5

2 Exponentials and Logarithms of Complex Numbers13

2.1 Raising e to Complex Powers13

2.2 Logarithms of Complex Numbers15

2.3 Raising Complex Numbers to Complex Powers16

3 Functions of a Complex Variable17

3.1 Complex Functions17

3.2 Combining Functions19

3.3 Functions as Mappings20

4 Exercises for Chapter Ⅰ25

Ⅱ The Rudiments of Plane Topology33

1 Basic Notation and Terminology33

1.1 Disks33

1.2 Interior Points,Open Sets34

1.3 Closed Sets34

1.4 Boundary,Closure,Interior35

1.5 Sequences35

1.6 Convergence of Complex Sequences36

1.7 Accumulation Points of Complex Sequences37

2 Continuity and Limits of Functions39

2.1 Continuity39

2.2 Limits of Functions43

3 Connected Sets47

3.1 Disconnected Sets47

3.2 Connected Sets48

3.3 Domains50

3.4 Components of Open Sets50

4 Compact Sets52

4.1 Bounded Sets and Sequences52

4.2 Cauchy Sequences53

4.3 Compact Sets54

4.4 Uniform Continuity57

5 Exercises for Chapter Ⅱ58

Ⅲ Analytic Functions62

1 Complex Derivatives62

1.1 Differentiability62

1.2 Differentiation Rules64

1.3 Analytic Functions67

2 The Cauchy-Riemann Equations68

2.1 The Cauchy-Riemann System of Equations68

2.2 Consequences of the Cauchy-Riemann Relations73

3 Exponential and Trigonometric Functions75

3.1 Entire Functions75

3.2 Trigonometric Functions77

3.3 The Principal Arcsine and Arctangent Functions81

4 Branches of Inverse Functions85

4.1 Branches of Inverse Functions85

4.2 Branches of the p th-root Function87

4.3 Branches of the Logarithm Function91

4.4 Branches of the λ-power Function92

5 Differentiability in the Real Sense96

5.1 Real Differentiability96

5.2 The Functions fz and f？98

6 Exercises for Chapter Ⅲ101

Ⅳ Complex Integration109

1 Paths in the Complex Plane109

1.1 Paths109

1.2 Smooth and Piecewise Smooth Paths112

1.3 Parametrizing Line Segments114

1.4 Reverse Paths,Path Sums115

1.5 Change of Parameter116

2 Integrals Along Paths118

2.1 Complex Line Integrals118

2.2 Properties of Contour Integrals122

2.3 Primitives125

2.4 Some Notation129

3 Rectifiable Paths131

3.1 Rectifiable Paths131

3.2 Integrals Along Rectifiable Paths133

4 Exercises for Chapter Ⅳ136

Ⅴ Cauchy's Theorem and its Consequences140

1 The Local Cauchy Theorem140

1.1 Cauchy's Theorem For Rectangles140

1.2 Integrals and Primitives144

1.3 The Local Cauchy Theorem148

2 Winding Numbers and the Local Cauchy Integral Formula153

2.1 Winding Numbers153

2.2 Oriented Paths,Jordan Contours160

2.3 The Local Integral Formula161

3 Consequences of the Local Cauchy Integral Formula164

3.1 Analyticity of Derivatives164

3.2 Derivative Estimates167

3.3 The Maximum Principle170

4 More About Logarithm and Power Functions175

4.1 Branches of Logarithms of Functions175

4.2 Logarithms of Rational Functions178

4.3 Branches of Powers of Functions182

5 The Global Cauchy Theorems185

5.1 Iterated Line Integrals185

5.2 Cycles186

5.3 Cauchy's Theorem and Integral Formula188

6 Simply Connected Domains194

6.1 Simply Connected Domains194

6.2 Simple Connectivity,Primitives,and Logarithms195

7 Homotopy and Winding Numbers197

7.1 Homotopic Paths197

7.2 Contractible Paths203

8 Exercises for Chapter Ⅴ204

Ⅵ Harmonic Functions214

1 Harmonic Functions215

1.1 Harmonic Conjugates215

2 The Mean Value Property219

2.1 The Mean Value Property219

2.2 Functions Harmonic in Annuli221

3 The Dirichlet Problem for a Disk226

3.1 A Heat Flow Problem226

3.2 Poisson Integrals228

4 Exercises for Chapter Ⅵ238

Ⅶ Sequences and Series of Analytic Functions243

1 Sequences of Functions243

1.1 Uniform Convergence243

1.2 Normal Convergence246

2 Infinite Series248

2.1 Complex Series248

2.2 Series of Functions253

3 Sequences and Series of Analytic Functions256

3.1 General Results256

3.2 Limit Superior of a Sequence259

3.3 Taylor Series260

3.4 Laurent Series269

4 Normal Families278

4.1 Normal Subfamilies of C(U)278

4.2 Equicontinuity279

4.3 The Arzelà-Ascoli and Montel Theorems282

5 Exercises for Chapter Ⅶ286

Ⅷ Isolated Singularities of Analytic Functions300

1 Zeros of Analytic Functions300

1.1 The Factor Theorem for Analytic Functions300

1.2 Multiplicity303

1.3 Discrete Sets,Discrete Mappings306

2 Isolated Singularities309

2.1 Definition and Classification of Isolated Singularities309

2.2 Removable Singularities310

2.3 Poles311

2.4 Meromorphic Functions318

2.5 Essential Singularities319

2.6 Isolated Singularities at Infinity322

3 The Residue Theorem and its Consequences323

3.1 The Residue Theorem323

3.2 Evaluating Integrals with the Residue Theorem326

3.3 Consequences of the Residue Theorem339

4 Function Theory on the Extended Plane349

4.1 The Extended Complex Plane349

4.2 The Extended Plane and Stereographic Projection350

4.3 Functions in the Extended Setting352

4.4 Topology in the Extended Plane354

4.5 Meromorphic Functions and the Extended Plane356

5 Exercises for Chapter Ⅷ362

Ⅸ Conformal Mapping374

1 Conformal Mappings375

1.1 Curvilinear Angles375

1.2 Diffeomorphisms377

1.3 Conformal Mappings379

1.4 Some Standard Conformal Mappings383

1.5 Self-Mappings of the Plane and Unit Disk388

1.6 Conformal Mappings in the Extended Plane389

2 M？bius Transformations391

2.1 Elementary M？bius Transformations391

2.2 M？bius Transformations and Matrices392

2.3 Fixed Points394

2.4 Cross-ratios396

2.5 Circles in the Extended Plane398

2.6 Reflection and Symmetry399

2.7 Classification of M？bius Transformations402

2.8 Invariant Circles408

3 Riemann's Mapping Theorem416

3.1 Preparations416

3.2 The Mapping Theorem419

4 The Carathéodory-Osgood Theorem423

4.1 Topological Preliminaries423

4.2 Double Integrals426

4.3 Conformal Modulus427

4.4 Extending Conformal Mappings of the Unit Disk440

4.5 Jordan Domains445

4.6 Oriented Boundaries447

5 Conformal Mappings onto Polygons450

5.1 Polygons450

5.2 The Reflection Principle451

5.3 The Schwarz-Christoffel Formula454

6 Exercises for Chapter Ⅸ466

Ⅹ Constructing Analytic Functions477

1 The Theorem of Mittag-Leffler477

1.1 Series of Meromorphic Functions477

1.2 Constructing Meromorphic Functions479

1.3 The Weierstrass ？-function486

2 The Theorem of Weierstrass490

2.1 Infinite Products490

2.2 Infinite Products of Functions493

2.3 Infinite Products and Analytic Functions495

2.4 The Gamma Function504

3 Analytic Continuation507

3.1 Extending Functions by Means of Taylor Series507

3.2 Analytic Continuation510

3.3 Analytic Continuation Along Paths512

3.4 Analytic Continuation and Homotopy517

3.5 Algebraic Function Elements520

3.6 Global Analytic Functions527

4 Exercises for Chapter Ⅹ535

Appendix A Background on Fields543

1 Fields543

1.1 The Field Axioms543

1.2 Subfields544

1.3 Isomorphic Fields544

2 Order in Fields545

2.1 Ordered Fields545

2.2 Complete Ordered Fields546

2.3 Implications for Real Sequences546

Appendix B Winding Numbers Revisited548

1 Technical Facts About Winding Numbers548

1.1 The Geometric Interpretation548

1.2 Winding Numbers and Jordan Curves550

Index556