Linear Differential Equations and Group Theory from Riemann to Poincare

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Springer Science & Business Media, 7 de gen. 2010 - 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)

 

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and his unpublished one on the hge 1812b 89
8
In his papers 1851 1857c
21
Monodromy matrices at the branchpoints and products of same
25
Lazarus Fuchs
41
dy Fp1x dHy F2plx Fnp1X
47
In the case of the elliptic integral he gave a thorough
53
Frobeniuss 1873 paper simplified the method of Fuchs
56
The solution is essentially that F a B y z is algebraic provided
72
Weierstrass 1853 1856 on the inversion of systems of hyperelliptic integrals
147
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

if a second order
85
Fuchs found his earlier solution contained invalid items
89
Brioschis calculation of hges corresponding to Schwarzs allowable
103
Galois on solvability by radicals
116
The rediscovery of his work in 1846 culminating in Jordans
117
The particular case when this surface is of genus zero
130
Pliicker 1834 on the 3n n 2 inflection points of a plane curve
144
Fuchss analysis of the nth order equation 247
253
PicardVessiot Theory
267
Notes
281
Chapter V
295
Historical List of Names
333
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