Translations of Mathematical Monographs 1992; 175 pp; softcover Volume: 96 ISBN10: 0821841645 ISBN13: 9780821841648 List Price: US$73 Member Price: US$58.40 Order Code: MMONO/96.S
 The theory of Abelian functions, which was at the center of nineteenthcentury mathematics, is again attracting attention. However, today it is frequently seen not just as a chapter of the general theory of functions but as an area of application of the ideas and methods of commutative algebra. This book presents an exposition of the fundamentals of the theory of Abelian functions based on the methods of the classical theory of functions. This theory includes the theory of elliptic functions as a special case. Among the topics covered are theta functions, Jacobians, and Picard varieties. The author has aimed the book primarily at intermediate and advanced graduate students, but it would also be accessible to the beginning graduate student or advanced undergraduate who has a solid background in functions of one complex variable. This book will prove especially useful to those who are not familiar with the analytic roots of the subject. In addition, the detailed historical introduction cultivates a deep understanding of the subject. Thorough and selfcontained, the book will provide readers with an excellent complement to the usual algebraic approach. Readership Upper level undergraduates, graduate students, and research mathematicians interested in analysis. Reviews "Readers ... will be grateful for this wellwritten modern account, which still preserves a historical perspective."  Bulletin of the London Mathematical Society "Very well suited for teaching and learning at the upper undergraduate level."  Zentralblatt MATH Table of Contents  Historical introduction. The Jacobian inversion problem
 Periodic functions of several complex variables
 Riemann matrices. Jacobian (intermediate) functions
 Construction of Jacobian functions of a given type. Theta functions and Abelian functions. Abelian and Picard manifolds
 Appendix A. Skewsymmetric determinants
 Appendix B. Divisors of analytic functions
 Appendix C. A summary of the most important formulas
