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Full Description
Complex analysis is one of the most central subjects in mathematics. It is compelling and rich in its own right, but it is also remarkably useful in a wide variety of other mathematical subjects, both pure and applied. This book is different from others in that it treats complex variables as a direct development from multivariable real calculus. As each new idea is introduced, it is related to the corresponding idea from real analysis and calculus. The text is rich with examples and exercises that illustrate this point. The authors have systematically separated the analysis from the topology, as can be seen in their proof of the Cauchy theorem. The book concludes with several chapters on special topics, including full treatments of special functions, the prime number theorem, and the Bergman kernel. The authors also treat $H^p$ spaces and Painleve's theorem on smoothness to the boundary for conformal maps. This book is a text for a first-year graduate course in complex analysis. It is an engaging and modern introduction to the subject, reflecting the authors' expertise both as mathematicians and as expositors.
Contents
Cover; Title page; Photos; Contents; Foreword; Preface; Permissions and acknowledgments; Curriculum vitae; List of former students; 44. Problem 9: The general reciprocity law; 45. Relations between 퐾₂ and Galois cohomology; 46. Local constants; 47. On the torsion in 퐾₂ of fields; 48. Fields medals (IV): An instinct for the key idea; 49. A simple proof of the main theorem of elimination theory in algebraic geometry; 50. Number theoretic background; 51. The Harish-Satake transform on 퐺퐿ᵣ; 52. Brumer-Stark-Stickelberger; 53. On conjugation of abelian varieties of CM type; 54. On Stark's conjectures on the behavior of 퐿(푠,휒) at 푠=0; 55. Variation of the canonical height of a point depending on a parameter; 56. A reciprocity law for 퐾₂-traces; 57. Canonical height pairings via biextensions; 58. On 푝-adic analogues of the conjectures of Birch and Swinnerton-Dyer; 59. Refined conjectures of the "Birch and Swinnerton-Dyer type"; 60. Commentary on algebra; 61. Some algebras associated to automorphisms of elliptic curves; 62. The 푝-adic sigma function; 63. Quantum deformations of 퐺퐿_{푛}; 64. Modules over regular algebras of dimension 3; 65. Conjectures on algebraic cycles in ℓ-adic cohomology; 66. The center of the 3-dimensional and 4-dimensional Sklyanin algebras; 67. The non-existence of certain Galois extensions of ℚ unramified outside 2; 68. The centers of 3-dimensional Sklyanin algebras; 69. A review of non-Archimedean elliptic functions; 70. Homological properties of Sklyanin algebras; 71. Linear forms in 푝-adic roots of unity; 72. Finite flat group schemes; 73. Bernard Dwork (1923-1998); 74. Galois cohomology; 75. On a conjecture of Finotti; 76. Refining Gross's conjecture on the values of abelian 퐿-functions; 77. On the Jacobians of plane cubics; 78. Computation of 푝-adic heights and log convergence; Letters; L19. Letter to Serre (1979/10/1); L20. Letter to Serre (1979/10/12); L21. Letter to Serre (1979/11/7); Back Cover
