Math8811: Complex Analysis
نویسنده
چکیده
1. Classical Topics 2 1.1. Complex numbers 2 1.2. Differentiability 2 1.3. Cauchy-Riemann Equations 3 1.4. The Riemann Sphere 4 1.5. Möbius transformations 5 1.6. Integration 6 1.7. Cauchy’s Theorems and Applications 7 1.8. Properties of Analytic Functions 9 1.9. The Winding Number 11 1.10. Singularities 12 1.11. Laurent Series 13 1.12. Residue Theory 15 1.13. Normal Families 18 1.14. The Riemann Mapping Theorem 19 1.15. Schwarz Reflection 21 1.16. Analytic Continuation 23 1.17. Analytic Covering Maps 24 1.18. The Picard Theorems 25 1.19. Infinite Products 27 2. Riemann Surfaces 34 2.1. Preliminaries 35 2.2. Sheaves 37 2.3. Maps between sheaves 38 2.4. Stalks and germs 39 2.5. Sheaf cohomology 40 2.6. Holomorphic vector bundles 45 2.7. Vector bundles and locally free sheaves 46 2.8. Divisors 46 2.9. Line bundles 48 2.10. Sections of a line bundle 50 2.11. The Riemann-Roch formula 52 2.12. The Riemann-Hurwitz formula 53 2.13. Genus formula of plane curves 55 2.14. Base point free and very ample line bundles 56
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تاریخ انتشار 2017