Complex analysis

Cover......Page 1
Preface to the First English Edition......Page 7
Contents......Page 9
Introduction......Page 13
1.1 Complex Numbers......Page 21
1.2 Convergent Sequences and Series......Page 36
1.3 Continuity......Page 48
1.4 Complex Derivatives......Page 54
1.5 The Cauchy-Riemann Differential Equations......Page 59
2 Integral Calculus in the Complex Plane C......Page 81
2.1 Complex Line Integrals......Page 82
2.2 The Cauchy Integral Theorem......Page 89
2.3 The Cauchy Integral Formulas......Page 104
3 Sequences and Series of Analytic Functions, the Residue Theorem......Page 115
3.1 Uniform Approximation......Page 116
3.2 Power Series......Page 121
3.3 Mapping Properties of Analytic Functions......Page 136
3.4 Singularities of Analytic Functions......Page 145
3.5 Laurent Decomposition......Page 154
A. Appendix to 3.4 and 3.5......Page 167
3.6 The Residue Theorem......Page 174
3.7 Applications of the Residue Theorem......Page 182
4 Construction of Analytic Functions......Page 203
4.1 The Gamma Function......Page 204
4.2 The Weierstrass Product Formula......Page 222
4.3 The Mittag-Leffler Partial Fraction Decomposition......Page 230
4.4 The Riemann Mapping Theorem......Page 235
A. Appendix : The Homotopical Version of the Cauchy Integral Theorem......Page 245
B. Appendix : A Homological Version of the Cauchy Integral Theorem......Page 251
C. Appendix : Characterizations of Elementary Domains......Page 256
5 Elliptic Functions......Page 263
5.1 Liouville's Theorems......Page 264
A. Appendix to the Definition of the Period Lattice......Page 271
5.2 The Weierstrass ρ-function......Page 273
5.3 The Field of Elliptic Functions......Page 279
A. Appendix to Sect. 5.3 : The Torus as an Algebraic Curve......Page 283
5.4 The Addition Theorem......Page 290
5.5 Elliptic Integrals......Page 296
5.6 Abel's Theorem......Page 303
5.7 The Elliptic Modular Group......Page 313
5.8 The Modular Function j......Page 321
6 Elliptic Modular Forms......Page 329
6.1 The Modular Group and Its Fundamental Region......Page 330
6.2 The k/12-formula and the Injectivity of the j-function......Page 338
6.3 The Algebra of Modular Forms......Page 346
6.4 Modular Forms and Theta Series......Page 350
6.5 Modular Forms for Congruence Groups......Page 364
A. Appendix to 6.5 : The Theta Group......Page 375
6.6 A Ring of Theta Functions......Page 382
7 Analytic Number Theory......Page 393
7.1 Sums of Four and Eight Squares......Page 394
7.2 Dirichlet Series......Page 411
7.3 Dirichlet Series with Functional Equations......Page 420
7.4 The Riemann ζ-function and Prime Numbers......Page 433
7.5 The Analytic Continuation of the ζ-function......Page 441
7.6 A Tauberian Theorem......Page 448
8.1 Solutions to the Exercises of Chapter I......Page 461
8.2 Solutions to the Exercises of Chapter 2......Page 471
8.3 Solutions to the Exercises of Chapter 3......Page 476
8.4 Solutions to the Exercises of Chapter 4......Page 487
8.5 Solutions to the Exercises of Chapter 5......Page 494
8.6 Solutions to the Exercises of Chapter 6......Page 502
8.7 Solutions to the Exercises of Chapter 7......Page 510
References......Page 521
Symbolic Notations......Page 531
Index......Page 533