مقدمه‌ای بر تحلیل مختلط: توابع چند متغیره: جلد دوم (ترجمه‌های مونوگراف‌های ریاضی)

Since the 1960s, there has been a flowering in higher-dimensional complex analysis. Both classical and new results in this area have found numerous applications in analysis, differential and algebraic geometry, and, in particular, contemporary mathematical physics. In many areas of modern mathematics, the mastery of the foundations of higher-dimensional complex analysis has become necessary for any specialist. Intended as a first study of higher-dimensional complex analysis, this book covers the theory of holomorphic functions of several complex variables, holomorphic mappings, and submanifolds of complex Euclidean space. Contents......Page 4 Foreword to the Third Edition......Page 8 1. The space C^n......Page 10 2. The simplest domains......Page 15 3. The concept of holomorphy......Page 21 4. Pluriharmonic functions......Page 24 5. Simplest properties of holomorphic functions......Page 27 6. The fundamental theorem of Hartogs......Page 33 7. Power series......Page 38 8. Other series......Page 42 9. Properties of holomorphic mappings......Page 48 10. Biholomorphic mappings......Page 53 11. Fatou's example......Page 63 Problems......Page 67 12. The concept of a manifold......Page 70 13. Complexification of Minkowski space......Page 76 14. Stokes's formula......Page 86 15. The Cauchy-Poincare theorem......Page 92 16. Maxwell's equations......Page 95 17.Submanifoldsof C^n......Page 105 18. Wirtinger's theorem......Page 110 19. The Fubini-Study form and related topics......Page 117 20. The concept of a covering......Page 121 21. Fundamental groups and coverings......Page 124 22. Riemann domains......Page 130 23. The Weierstrass preparation theorem......Page 132 24. Properties of analytic sets......Page 138 25. Local structure......Page 145 26. The concept of a bundle......Page 149 27. The tangent and cotangent bundles......Page 152 28. The concept of a sheaf......Page 157 Problems......Page 161 29. The formulas of Martinelli-Bochner and Leray......Page 164 30. Weil's formula......Page 170 31. Extension from the boundary......Page 175 32. Hartogs's theorem and the removal of singularities......Page 182 33. The concept of a domain of holomorphy......Page 186 34. Holomorphic convexity......Page 190 35. Properties of domains of holomorphy......Page 194 36. The continuity principle......Page 197 37. Local pseudoconvexity......Page 201 38. Plurisubharmonic functions......Page 208 39. Pseudoconvex domains......Page 215 40. One-sheeted envelopes......Page 221 41. Multiple-sheeted envelopes......Page 226 42. Analyticity of the set of singularities......Page 232 Problems......Page 237 43. The concept of a meromorphic function......Page 240 44. The first Cousin problem......Page 243 45. Solution of the first problem......Page 247 46. Cohomology groups......Page 251 47. Exact sequences of sheaves......Page 256 48. Localized first Cousin problem......Page 258 49. Second Cousin problem......Page 262 50. Applications of the Cousin problems......Page 268 51. Solution of the Levi problem......Page 271 52. Other applications......Page 273 53. The theory of Martinelli......Page 281 54. The theory of Leray......Page 287 55. Logarithmic residue......Page 295 Problems......Page 301 56. The Bergman metric......Page 304 57. The Caratheodory metric......Page 311 58. The Kobayashi metric......Page 314 59. Criteria for hyperbolicity......Page 317 60. Generalizations of Picard's theorem......Page 326 61. Mappings of strictly pseudoconvex domains......Page 336 62. Correspondence of boundaries......Page 341 63. A symmetry principle......Page 345 64. Vector fields......Page 350 65. Boundary properties of functions......Page 356 66. Uniqueness theorems and propositions......Page 361 Problems......Page 368 Appendix. Complex Potential Theory......Page 370 Subject Index......Page 378 Pt. 2. Functions Of Several Variables. B.v. Shabat ; [translated From The Russian By J.s. Joel].