Linear differential equations and group theory from Riemann to Poincaré
著者
書誌事項
Linear differential equations and group theory from Riemann to Poincaré
Birkhäuser, c2000
2nd ed.
- : us
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注記
Includes bibliographical references (p. [307]-334) and index
内容説明・目次
- 巻冊次
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: us ISBN 9780817638375
内容説明
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. The exercises have been retained, making it possible to use the book as a companion to mathematics courses at the graduate level.
目次
Hypergeometric Equations.- Lazarus Fuchs.- Algebraic Solutions to a Differential Equation.- Modular Equations.- Some Algebraic Curves.- Automorphic Functions.
- 巻冊次
-
ISBN 9783764338374
内容説明
This is a study of how a particular version of the unity of mathematics, often called geometric function theory, was created in the 19th century. The focus is on three mathematical topics: hypergeometric and related linear differential equations, group theory, and non-Euclidean geometry.
目次
- Hypergeometric equations and modular equations - Euler and Gauss, Jacobi and Kummer, Riemann's approach to complex analysis, Riemann's P-function, interlude -Cauchy's theory of differential equations
- Lazarus Fuchs - Fuchs's theory of linear equations, generalisation of the hypergeometric equation, conclusion, the new methods of Frobenius and other
- algebraic solutions to a differential equation -Scharz, generalisations, Klein and Gordan, the solutions of Gordan and Fuchs, Jordan's solution, equations of higher order
- modular equations - Fuchs and Hermite, Dedekind, Galois theory, groups and fields, the Galois theory of module equations, c.1858, Klein
- some algebraic curves - algebraic curves, particularly quartics, function-theoretic geometry, Klein
- automorphic functions - Lame's equation, Poincare, Klein, 1881, Klein's response, Poincare's papers of 1883 and 1884. Appendices: Riemann, Schottky, and Schwarz on conformal representation
- Riemann's lectures and the Riemann-Hilbert problem
- Fuchs's analysis of the nth order equation
- on the history of non-Euclidean geometry
- the uniformisation theorem
- Picard-Vessiot theory
- the hypergeometric equation in several variables - Appell and Picard. Notes on chapters and appendices. Bibliography.
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