| 书目名称 | Linear Differential Equations and Group Theory from Riemann to Poincare | | 编辑 | Jeremy J. Gray | | 视频video | http://file.papertrans.cn/587/586302/586302.mp4 | | 概述 | Chronicles important events not covered anywhere else.Discusses the history and development of ideas, not just mathematicians | | 丛书名称 | Modern Birkhäuser Classics | | 图书封面 |  | | 描述 | .This book is a study of how a particular vision of the unity of mathematics, often called geometric function theory, was created in the 19th century. The central focus is on the convergence of three mathematical topics: the hypergeometric and related linear differential equations, group theory, and on-Euclidean geometry...The text for this second edition has been greatly expanded and revised, and the existing appendices enriched with historical accounts of the Riemann–Hilbert problem, the uniformization theorem, Picard–Vessiot theory, and the hypergeometric equation in higher dimensions. The exercises have been retained, making it possible to use the book as a companion to mathematics courses at the graduate level.. | | 出版日期 | Textbook 2008Latest edition | | 关键词 | Algebra; Equations; Felix Klein; Group theory; History; History of Mathematics; Lazarus Fuchs; Poincaré; Rie | | 版次 | 2 | | doi | https://doi.org/10.1007/978-0-8176-4773-5 | | isbn_softcover | 978-0-8176-4772-8 | | isbn_ebook | 978-0-8176-4773-5Series ISSN 2197-1803 Series E-ISSN 2197-1811 | | issn_series | 2197-1803 | | copyright | Birkhäuser Boston 2008 |
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Front Matter |
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Abstract
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,Hypergeometric Equations, |
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Abstract
This chapter does three things. It gives a short account of the work of Euler, Gauss, Kummer, and Riemann on the hypergeometric equation, with some indication of its immediate antecedents and consequences. It therefore looks very briefly at some of the work of Gauss, Legendre, Abel and Jacobi on elliptic functions, in particular at their work on modular functions and modular transformations. It concludes with a description of the general theory of linear differential equations supplied by Cauchy and Weierstrass. There are many omissions, some of which are rectified elsewhere in the literature.. The sole aim of this chapter is to provide a setting for the work of Fuchs on linear ordinary differential equations, to be discussed in Chapter II, and for later work on modular functions, discussed in Chapters IV and V.
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,Lazarus Fuchs, |
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Abstract
In the years 1865, 1866, and 1868, Lazarus Fuchs published three papers, each entitled “Zur Theorie der Linearen Differentialgleichungen mit veränderlichen Coefficienten” (“On the theory of linear differential equations with variable coefficients”). These will be surveyed in this chapter. In them he characterised the class of linear differential equations in a complex variable ., all of whose solutions have only finite poles and possibly logarithmic branch points. So, near any point . in the domain of the coefficients, the solutions become finite and singled-valued upon multiplication by a suitable power of (. − .) unless it involves a logarithmic term. This class came to be called the Fuchsian class, and equations in it, equations of the Fuchsian type. As will be seen, it contains many interesting equations, including the hypergeometric. In the course of this work, Fuchs created much of the elementary theory of linear differential equations in the complex domain: the analysis of singular points; the nature of a basis of . linearly independent solutions to an equation of degree . when there are repeated roots of the indicial equation; explicit forms for the solution according to th
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,Algebraic Solutions to a Differential Equation, |
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Abstract
This chapter considers how Fuchs’ problem: when are all solutions to a linear ordinary differential equation algebraic? was approached, and solved, in the 1870s and 1880s. First, Schwarz solved the problem for the hypergeometric equation. Then Fuchs solved it for the general second-order equation by reducing it to a problem in invariant theory and solving that problem by . means. Gordan later solved the invariant theory problem directly. But Fuchs’ solution was imperfect, and Klein simplified and corrected it by a mixture of geometric and group-theoretic techniques which established the central role played by the regular solids already highlighted by Schwarz. Simultaneously, Jordan showed how the problem could be solved by purely group-theoretic means, by reducing it to a search for all finite monodromy groups of 2 × 2 matrices with complex entries and determinant 1. He was also able to solve it for 3rd and 4th order equations, thus providing the first successful treatment of the higher order cases, and to prove a general finiteness theorem for the .th order case (Jordan’s finiteness theorem). Later Fuchs and Halphen were able to treat some of these cases invariant-theoretically.
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,Modular Equations, |
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Abstract
The study of the algebraic-solutions problem for a second-order linear ordinary differential equation had brought to light the conceptual importance of considering groups of motions of the sphere, and, in particular, finite groups. Klein connected this study with that of the quintic equation, and so with the theory of transformations of elliptic functions and modular equations as considered by Hermite, Brioschi, and Kronecker around 1858. Klein’s approach to the modular equations was first to obtain a better understanding of the moduli, and this led him to the study of the upper half plane under the action of the group of two-by-two matrices with integer entries and determinant one; his great achievement was the production of a unified theory of modular functions. Independently of him, Dedekind also investigated these questions from the same standpoint, in response to a paper of Fuchs. So this chapter looks first at Fuchs’ study of elliptic integrals as a function of a parameter, and then at the work of Dedekind. The algebraic study of the modular equation is then discussed; the chapter concludes with Klein’s unification of these ideas.
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,Some Algebraic Curves, |
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Abstract
This chapter discusses a topic which was studied from various points of view throughout the nineteenth century and which presented itself in such different guises as: the 28 bitangents to a quartic curve, the study of a Riemann surface of genus 3 and its group of automorphisms, and the reduction of the modular equation of degree 8. These studies, which began separately, were drawn together by Klein in 1878 and proved crucial to his discovery of automorphic functions.
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,Automorphic Functions, |
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Abstract
This final chapter is, naturally, concerned with the triumphant accomplishments of Poincaré: the creation of the theory of Fuchsian groups and automorphic functions. These developments brought together the theory of linear differential equations and the group-theoretic approach to the study of Riemann surfaces, so this account draws on all of the preceding material. It begins with a significant stage that is intermediate between the embryonic general theory and the developed Fuchsian theory: Lamé’s equation.
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Back Matter |
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Abstract
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