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小平邦彦复分析2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

小平邦彦著著
出版社：人民邮电出版社
ISBN：7115178404
出版时间：2008
标注页数：404页
文件大小：96MB
文件页数：40213831页
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图书目录

1 Holomorphic functions1

1.1 Holomorphic functions1

a.The complex plane1

b.Functions of a complex variable5

c.Holomorphic functions9

1.2 Power series15

a.Series whose terms are functions15

b.Power series16

1.3 Integrals24

a.Curves25

b.Integrals30

c.Cauchy’s integral formula for circles35

d.Power series expansions43

1.4 Properties of holomorphic functions47

a.mth-order derivatives47

b.Limits of sequences of holomorphic functions49

c.The Mean Value Theorem and the maximum principle51

d.Isolated singularities52

e.Entire functions58

2 Cauchy’s Theorem60

2.1 Piecewise smooth curves60

a.Smooth Jordan curves60

b.Boundaries of bounded closed regions65

2.2 Cellular decomposition74

a.Calls74

b.Cellular decomposition80

2.3 Cauchy’s Theorem101

a.Cauchy’s Theorem101

b.Cauchy’s integral formula104

c.Residues106

d.Evaluation of definite integrals109

2.4 Differentiability and homology113

3 Conformal mappings117

3.1 Conformal mappings117

3.2 The Riemann sphere132

a.The Riemann sphere132

b.Holomorphic functions with an isolated singularity at ∞135

c.Local coordinates137

d.Homogeneous coordinates138

3.3 Linear fractional transformations139

a.Linear fractional transformations139

b.Cross ratio142

c.Elementary conformal mappings148

4 Analytic continuation153

4.1 Analytic continuation153

a.Analytic continuation153

b.Analytic continuation by expansion in power series157

4.2 Analytic continuation along curves160

4.3 Analytic continuation by integrals180

4.4 Cauchy’s Theorem （continued）190

5 Riemann’s Mapping Theorem200

5.1 Riemann’s Mapping Theorem200

5.2 Correspondence of boundaries214

5.3 The principle of reflection224

a.The principle of reflection224

b.Modular functions233

c.Picard’s Theorem241

d.The Schwarz-Christoffel formula241

6 Riemann surfaces247

6.1 Differential forms247

a.Differential forms247

b.Line integrals249

c.Harmonic forms257

d.Harmonic functions258

6.2 Riemann surfaces260

a.Hausdorff spaces260

b.Definition of Riemann surfaces263

6.3 Differential forms on a Riemann surface268

a.Differential forms268

b.Line integrals272

c.Locally finite open coverings275

d.Partition of unity278

e.Green’s Theorem284

6.4 Dirichlet’s Principle294

a.Inner product and norm294

b.Dirichlet’s Principle299

c.Analytic functions314

7 The structure of Riemann surfaces319

7.1 Planar Riemann surfaces319

a.Planar Riemann surfaces319

b.Simply connected Riemann surfaces329

c.Multiply connected regions333

7.2 Compact Riemann surfaces340

a.Cohomology groups340

b.Structure of compact Riemann surfaces344

c.Homology groups360

8 Analytic functions on a closed Riemann surface376

8.1 Abelian differentials of the first kind376

a.Harmonic 1-forms of the first kind376

b.Abelian differential of the first kind379

8.2 Abelian differentials of the second and third kind379

a.Meromorphic functions379

b.Abelian differentials of the second and third kind380

8.3 The Riemann-Roch Theorem381

a.Existence Theorem381

b.The Riemann-Roch Theorem382

8.4 Abel’s Theorem389

a.Existence Theorem389

b.Abel’s Theorem391

Problems394

Index401