Preface Chapter 1 Covering Spaces 1. The Definition of Riemann Surfaces 2. Elementary Properties of Holomorphic Mappings 3. Homotopy of Curves. The Fundamental Group 4. Branched and Unbranched Coverings 5. The Universal Covering and Covering Transformations 6. Sheaves 7. Analytic Continuation 8. Algebraic Functions 9. Differential Forms 10. The Integration of Differential Forms 11. Linear Differential Equations Chapter 2 Compact Riemann Surfaces 12. Cohomology Groups 13. Dolbeault''s Lemma 14. A Finiteness Theorem 15. The Exact Cohomology Sequence 16. The Riemann-Roch Theorem 17. The Serre Duality Theorem 18. Functions and Differential Forms with Prescribed Principal Parts 19. Harmonic Differential Forms 20. Abel''s Theorem 21. The Jacobi Inversion Problem Chapter 3 Non-compact Riemann Surfaces 22. The Dirichlet Boundary Value Problem 23. Countable Topology 24. Weyl's Lemma 25. The Runge Approximation Theorem 26. The Theorems of Mittag-Leffler and Weierstrass 27. The Riemann Mapping Theorem 28. Functions with Prescribed Summands of Automorphy 29. Line and Vector Bundles 30. The Triviality of Vector Bundles 31. The Riemann-Hilbert Problem Appendix A. Partitions of Unity B. Topological Vector Spaces References Symbol Index Author and Subject Index