Chapter 1 Covering Spaces 1
1．The Definition of Riemann Surfaces 1
2．Elementary Properties of Holomorphie Mappings 10
3．Homotopy of Curves.The Fundamental Group 13
4．Branched and Unbranched Coverings 20
5．The Universal Covering and Covering Transformations 31
6．Sheaves 40
7．Analytic Continuation 44
8．Algebraic Functions 48
9．Differential Forms 59
10．The Integration of Differential Forms 68
11．Linear Differential Equations 81
Chapter 2 Compact Riemann Surfaces 96
12．Cohomology Groups 96
13．Dolbeault's Lemma 104
14．A Finiteness Theorem 109
15．The Exact Cohomology Sequence 118
16．The Riemann-Roch Theorem 126
17．The Serre Duality Theorem 132
18．Functions and Differential Forms with Prescribed Principal Parts 146
19．Harmonic Differential Forms 153
20．Abel's Theorem 159
21．The Jacobi Inversion Problem 166
Chapter 3 Non-compact Riemann Surfaces 175
22．The Dirichlet Boundary Value Problem 175
23．Countable Topology 185
24．Weyl's Lemma 190
25．The Runge Approximation Theorem 196
26．The Theorems of Mittag-Leffler and Weierstrass 201
27．The Riemann Mapping Theorem 206
28．Functions with Prescribed Summands of Automorphy 214
29．Line and Vector Bundles 219
30．The Triviality of Vector Bundles 228
31．The Riemann-Hilbert Problem 231
Appendix 237
A．Partitions of Unity 237
B．Topological Vector Spaces 238
References 243
Symbol Index 247
Author and Subject Index 249