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Ⅰ Differential Calculus in the Complex Plane C 9

Ⅰ.1 Complex Numbers 9

Ⅰ.2 Convergent Sequences and Series 24

Ⅰ.3 Continuity 36

Ⅰ.4 Complex Derivatives 42

Ⅰ.5 The CAUCHY-RIEMANN Differential Equations 47

Ⅱ Integral Calculus in the Complex Plane C 69

Ⅱ.1 Complex Line Integrals 70

Ⅱ.2 The CAUCHY Integral Theorem 77

Ⅱ.3 The CAUCHY Integral Formulas 92

Ⅲ Sequences and Series of Analytic Functions,the Residue Theorem 103

Ⅲ.1 Uniform Approximation 104

Ⅲ.2 Power Series 109

Ⅲ.3 Mapping Properties of Analytic Functions 124

Ⅲ.4 Singularities of Analytic Functions 133

Ⅲ.5 LAURENT Decomposition 142

A Appendix to Ⅲ.4 and Ⅲ.5 155

Ⅲ.6 The Residue Theorem 162

Ⅲ.7 Applications of the Residue Theorem 170

Ⅳ Construction of Analytic Functions 191

Ⅳ.1 The Gamma Function 192

Ⅳ.2 The WEIERSTRASS Product Formula 210

Ⅳ.3 The MITTAG-LEFFLER Partial Fraction Decomposition 218

Ⅳ.4 The RIEMANN Mapping Theorem 223

A Appendix:The Homotopieal Version of the CAUCHY Integral Theorem 233

B Appendix:A Homological Version of the CAUCHY Integral Theorem 239

C Appendix:Characterizations of Elementary Domains 244

Ⅴ Elliptic Functions 251

Ⅴ.1 LIOUVILLE's Theorems 252

A Appendix to the Definition of the Period Lattice 259

Ⅴ.2 The WEIERSTRASS ？-function 261

Ⅴ.3 The Field of Elliptic Functions 267

A Appendix to Sect.Ⅴ.3:The Torus as an Algebraic Curve 271

Ⅴ.4 The Addition Theorem 278

Ⅴ.5 Elliptic Integrals 284

Ⅴ.6 ABEL's Theorem 291

Ⅴ.7 The Elliptic Modular Group 301

Ⅴ.8 The Modular Function j 309

Ⅵ Elliptic Modular Forms 317

Ⅵ.1 The Modular Group and Its Fundamental Region 318

Ⅵ.2 The k/12-formula and the Injectivity of the j-function 326

Ⅵ.3 The Algebra of Modular Forms 334

Ⅵ.4 Modular Forms and Theta Series 338

Ⅵ.5 Modular Forms for Congruence Groups 352

A Appendix to Ⅵ.5:The Theta Group 363

Ⅵ.6 A Ring of Theta Functions 370

Ⅶ Analytic Number Theory 381

Ⅶ.1 Sums of Four and Eight Squares 382

Ⅶ.2 DIRICHLET Series 399

Ⅶ.3 DIRICHLET Series with Functional Equations 408

Ⅶ.4 The RIEMANN ζ-function and Prime Numbers 421

Ⅶ.5 The Analytic Continuation of the ζ-function 429

Ⅶ.6 A TAUBERian Theorem 436

Ⅷ Solutions to the Exercises 449

Ⅷ.1 Solutions to the Exercises of Chapter Ⅰ 449

Ⅷ.2 Solutions to the Exercises of Chapter Ⅱ 459

Ⅷ.3 Solutions to the Exercises of Chapter Ⅲ 464

Ⅷ.4 Solutions to the Exercises of Chapter Ⅳ 475

Ⅷ.5 Solutions to the Exercises of Chapter Ⅴ 482

Ⅷ.6 Solutions to the Exercises of Chapter Ⅵ 490

Ⅷ.7 Solutions to the Exercises of Chapter Ⅶ 498

References 509

Symbolic Notations 519