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复分析PDF|Epub|txt|kindle电子书版本网盘下载

（美）加默兰著著
出版社：世界图书出版公司北京公司
ISBN：7506292297
出版时间：2008
标注页数：478页
文件大小：100MB
文件页数：40227501页
主题词：复分析－英文

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图书目录

FIRST PART1

Chapter Ⅰ The Complex Plane and Elementary Functions1

1. Complex Numbers1

2. Polar Representation5

3. Stereographic Projection11

4. The Square and Square Root Functions15

5. The Exponential Function19

6. The Logarithm Function21

7. Power Functions and Phase Factors24

8. Trigonometric and Hyperbolic Functions29

Chapter Ⅱ Analytic Functions33

1. Review of Basic Analysis33

2. Analytic Functions42

3. The Cauchy-Riemann Equations46

4. Inverse Mappings and the Jacobian51

5. Harmonic Functions54

6. Conformal Mappings58

7. Fractional Linear Transformations63

Chapter Ⅲ Line Integrals and Harmonic Functions70

1. Line Integrals and Green's Theorem70

2. Independence of Path76

3. Harmonic Conjugates83

4. The Mean Value Property85

5. The Maximum Principle87

6. Applications to Fluid Dynamics90

7. Other Applications to Physics97

Chapter Ⅳ Complex Integration and Analyticity102

1. Complex Line Integrals102

2. Fundamental Theorem of Calculus for Analytic Functions107

3. Cauchy's Theorem110

4. The Cauchy Integral Formula113

5. Liouville's Theorem117

6. Morera's Theorem119

7. Goursat's Theorem123

8. Complex Notation and Pompeiu's Formula124

Chapter Ⅴ Power Series130

1. Infinite Series130

2. Sequences and Series of Functions133

3. Power Series138

4. Power Series Expansion of an Analytic Function144

5. Power Series Expansion at Infinity149

6. Manipulation of Power Series151

7. The Zeros of an Analytic Function154

8. Analytic Continuation158

Chapter Ⅵ Laurent Series and Isolated Singularities165

1. The Laurent Decomposition165

2. Isolated Singularities of an Analytic Function171

3. Isolated Singularity at Infinity178

4. Partial Fractions Decomposition179

5. Periodic Functions182

6. Fourier Series186

Chapter Ⅶ The Residue Calculus195

1. The Residue Theorem195

2. Integrals Featuring Rational Functions199

3. Integrals of Trigonometric Functions203

4. Integrands with Branch Points206

5. Fractional Residues209

6. Principal Values212

7. Jordan's Lemma216

8. Exterior Domains219

SECOND PART224

Chapter Ⅷ The Logarithmic Integral224

1. The Argument Principle224

2. Rouche's Theorem229

3. Hurwitz's Theorem231

4. Open Mapping and Inverse Function Theorems232

5. Critical Points236

6. Winding Numbers242

7. The Jump Theorem for Cauchy Integrals246

8. Simply Connected Domains252

Chapter Ⅸ The Schwarz Lemma and Hyperbolic Geometry260

1. The Schwarz Lemma260

2. Conformal Self-Maps of the Unit Disk263

3. Hyperbolic Geometry266

Chapter Ⅹ Harmonic Functions and the Reflection Principle274

1. The Poisson Integral Formula274

2. Characterization of Harmonic Functions280

3. The Schwarz Reflection Principle282

Chapter Ⅺ Conformal Mapping289

1. Mappings to the Unit Disk and Upper Half-Plane289

2. The Riemann Mapping Theorem294

3. The Schwarz-Christoffel Formula296

4. Return to Fluid Dynamics304

5. Compactness of Families of Functions306

6. Proof of the Riemann Mapping Theorem311

THIRD PART315

Chapter Ⅻ Compact Families of Meromorphic Functions315

1. Marty's Theorem315

2. Theorems of Montel and Picard320

3. Julia Sets324

4. Connectedness of Julia Sets333

5. The Mandelbrot Set338

Chapter ⅩⅢ Approximation Theorems342

1. Runge's Theorem342

2. The Mittag-Leffler Theorem348

3. Infinite Products352

4. The Weierstrass Product Theorem358

Chapter ⅩⅣ Some Special Functions361

1. The Gamma Function361

2. Laplace Transforms365

3. The Zeta Function370

4. Dirichlet Series376

5. The Prime Number Theorem382

Chapter ⅩⅤ The Dirichlet Problem390

1. Green's Formulae390

2. Subharmonic Functions394

3. Compactness of Families of Harmonic Functions398

4. The Perron Method402

5. The Riemann Mapping Theorem Revisited406

6. Green's Function for Domains with Analytic Boundary407

7. Green's Function for General Domains413

Chapter ⅩⅥ Riemann Surfaces418

1. Abstract Riemann Surfaces418

2. Harmonic Functions on a Riemann Surface426

3. Green's Function of a Surface429

4. Symmetry of Green's Function434

5. Bipolar Green's Function436

6. The Uniformization Theorem438

7. Covering Surfaces441

Hints and Solutions for Selected Exercises447

References469

List of Symbols471

Index473