Linear Differential Equations and Group Theory from Riemann to PoincareSpringer Science & Business Media, 21 de gen. 2008 - 338 pàgines 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. "If you want to know what mathematicians like Gauss, Euler and Dirichlet were doing...this book could be for you. It fills in many historical gaps, in a story which is largely unknown...This book is the result of work done by a serious historian of mathematics...If you are intrigued by such topics studied years ago but now largely forgotten...then read this book."--The Mathematical Gazette (on the second edition) "One among the most interesting books on the history of mathematics... Very stimulating reading for both historians of modern mathematics and mathematicians as well."--Mathematical Reviews (on the first edition) "The book contains an amazing wealth of material relating to the algebra, geometry, and analysis of the nineteenth century.... Written with accurate historical perspective and clear exposition, this book is truly hard to put down."--Zentralblatt fur Mathematik (review of 1st edition) |
Continguts
so the hge is characterised by its exponents at the branchpoints | 1 |
2 | 7 |
and his unpublished one on the hge 1812b | 8 |
Cauchys proofs of the existence of solutions to differential equations | 19 |
In his papers 1851 1857c | 21 |
Lazarus Fuchs | 41 |
Frobeniuss 1873 paper simplified the method of Fuchs | 55 |
d | 62 |
Methodological questions | 161 |
Correspondence with Fuchs the birth of Fuchsian functions | 174 |
Poincarés first papers Fuchsian groups thetaFuchsian functions | 186 |
The discovery of the Grenzkreis theorem | 199 |
His work in 1880 1881 185 | 203 |
the cases of 4 and 5 singular points | 212 |
Riemanns lectures and the RiemannHilbert problem | 231 |
Fuchss analysis of the nth order equation | 247 |
The introduction of Jordan canonical form by Jordan and Hamburger | 63 |
the upper plane onto a triangle whose reflections tessellate the sphere | 72 |
Jordans main paper 1878 looks for finite subgroups | 75 |
Klein and Gordan | 81 |
Chapter | 101 |
Hermite on a certain matrix group | 107 |
Dedekinds approach based on the latticeidea and invariance under | 115 |
The rediscovery of his work in 1846 culminating in Jordans | 117 |
The connection with invariant theory | 125 |
The particular case when this surface is of genus zero | 130 |
Galois on solvability by radicals | 138 |
Plücker 1834 on the 3nn 2 inflection points of a plane curve | 144 |
Weierstrass 1853 1856 on the inversion of systems of hyperelliptic integrals | 147 |
polynomial in x of degree at most | 250 |
107 | 255 |
The uniformisation theorem | 261 |
The hypergeometric equation in several variables | 275 |
Notes | 281 |
Chapter V | 295 |
77 | 300 |
83 | 310 |
Such equations are of the Fuchsian class | 311 |
Fuchs gave rigorous methods for solving linear differential equations | 326 |
330 | |
Amongst the equations of the Fuchsian class are those all of whose solutions | 333 |
336 | |
Altres edicions - Mostra-ho tot
Linear Differential Equations and Group Theory from Riemann to Poincare Jeremy Gray Previsualització limitada - 2010 |
Differential Equations and Group Theory from Riemann to Poincare GRAY Visualització de fragments - 1986 |
Linear Differential Equations and Group Theory from Riemann to Poincare Jeremy Gray Previsualització no disponible - 2008 |