مقدمه‌ای بر نظریه کلاسیک توابع ابلیان

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. Readership: Upper level undergraduates, graduate students, and research mathematicians interested in analysis. Cover S Title Titles in This Series Introduction to the Classical Theory of Abelian Functions Copyright ©1992 by the American Mathematical Society. ISBN 0-8218-4542-X QA345.M4513 1992 515-dc20 LCCN 91-36838 CIP Contents Preface CHAPTER I Historical Introduction. The Jacobi Inversion Problem §1. Euler's equation §2. Elliptic functions of Gauss §3. Jacobi's inversion method §4. Jacobi identities §5. Problem of inversion of a hyperelliptic integral §6. Problem of Jacobi. Gopel and Rosenhain §7. Algebraic functions and their Riemann surfaces §8. Abelian integrals. Abel's theorem §9. Main directions of development of the theory of Abelian functions CHAPTER II Periodic Functions of Several Complex Variables §1. Divisibility relation for functions analytic at a point §2. Entire and meromorphic functions §3. The set of periods. Infinitesimal periods §4. Conditions for independence of periods §5. The fundamental system of periods §6. Transformation of period matrix §7. Generalized Fourier series of a periodic entire function §8. Construction of an entire function of one variable from its difference CHAPTER III Riemann Matrices. Jacobian (Intermediate) Functions §1. Riemann matrices. Elementary conditions §2. The first system of difference equations. Conditions of solvability §3. Construction of the solution of the first system §4. Jacobian or intermediate functions. The second system of difference equations §5. Solvability conditions for the second system and the solution of the second system §6. First and second period matrices. The characteristic matrix N §7. Upper bound on absolute values of Jacobian functions. Riemann inequality §8. Statement of necessary and sufficient conditions for a Riemann matrix. The principal matrix CHAPTER IV Construction of Jacobian Functions of a Given Type. Theta Functions and Abelian Functions. Abelian and Picard Manifolds §1. Construction of the characteristic matrix N and the second period matrix A, using given \Omega and P §2. Construction of Jacobian functions of a given type. Theta functions §3. Construction of Abelian functions. Fields of Abelian functions §4. Theta functions with characteristics §5. Weierstrass-Poincare theorem §6. Abelian and Picard manifolds §7. Rational functions on Picard varieties. Picard integrals of the first kind(24) Appendix §A. Skew-Symmetric Determinants A.1. The Pfafl'ian A.2. The Frobenius Theorem. §B. Divisors of analytic functions B.1. General theorems B.2. Continuation of the divisibility relation from a point to a region. B.3. Poincare-Cousin theorem §C. A summary of the most important formulas Jacobi function The Weierstrass preparation theorem. Reduced representations condition for the degeneracy of a function Period matrices Fourier series expansion Construction of an entire function from its difference. Riemann matrices. Elementary conditions. The first system of difference equations. Jacobian functions Necessary and sufficient conditions for \Omega Theta function Algebraic dependencies Back Cover Lie groups are very general mathematical objects that appear in numerous areas such as topology, functional analysis, and algebra, as well as differential geometry and differential topology. This book provides a guide to the topology of Lie groups and homogeneous spaces by bringing together a wide range of results relating to them. Presents an exposition of the fundamentals of the theory of Abelian functions based on the methods of the classical theory of functions. This book covers such topics as theta functions, Jacobians, and Picard varieties. A.i. Markushevich ; [translated From The Russian By G. Bluher ; Translation Edited By Ralph P. Boas]. Translation Of: Vvedenie V Klassicheskui︠u︡ Teorii︠u︡ Abelevykh Funkt︠s︡iĭ.