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Introduction to the Classical Theory of Abelian Functions
A. Markushevich

Translations of Mathematical Monographs
1992; 175 pp; softcover
Volume: 96
ISBN-10: 0-8218-4164-5
ISBN-13: 978-0-8218-4164-8
List Price: US$77
Member Price: US$61.60
Order Code: MMONO/96.S
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The theory of Abelian functions, which was at the center of nineteenth-century 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 self-contained, the book will provide readers with an excellent complement to the usual algebraic approach.


Upper level undergraduates, graduate students, and research mathematicians interested in analysis.


"Readers ... will be grateful for this well-written 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. Skew-symmetric determinants
  • Appendix B. Divisors of analytic functions
  • Appendix C. A summary of the most important formulas
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