Linear Differential Equations and Group Theory from Riemann by Jeremy Gray

By Jeremy Gray

This ebook is a examine of the way a selected imaginative and prescient of the team spirit of arithmetic, referred to as geometric functionality conception, used to be created within the 19th century. The relevant concentration is at the convergence of 3 mathematical subject matters: the hypergeometric and similar linear differential equations, crew conception, and on-Euclidean geometry.

The textual content for this moment variation has been vastly elevated and revised, and the prevailing appendices enriched with old money owed of the Riemann–Hilbert challenge, the uniformization theorem, Picard–Vessiot thought, and the hypergeometric equation in better dimensions. The routines were retained, making it attainable to take advantage of the e-book as a significant other to arithmetic classes on the graduate level.

"If you must recognize what mathematicians like Gauss, Euler and Dirichlet have been doing...this booklet can be for you. It fills in lots of ancient gaps, in a narrative that is principally unknown...This e-book is the results of paintings performed by way of a significant historian of mathematics...If you're intrigued through such issues studied years in the past yet now mostly forgotten...then learn this book."--The Mathematical Gazette (on the second one edition)

"One one of the best books at the heritage of mathematics... Very stimulating interpreting for either historians of contemporary arithmetic and mathematicians as well."--Mathematical Reviews (on the 1st edition)

"The e-book includes an grand wealth of fabric on the subject of the algebra, geometry, and research of the 19th century.... Written with actual historic standpoint and transparent exposition, this booklet is actually challenging to place down."--Zentralblatt fur Mathematik (review of 1st edition)

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Extra resources for Linear Differential Equations and Group Theory from Riemann to Poincaré

Example text

These are Gauss' contiguous functions transformed to Riemann's setting. In this way he could determine which differential equation the P-function satisfies, since, dy d2y P = y , P I = dx and / ~ dx 2 are three such P-functions and, Riemann showed, they satisfy a linear relationship with certain rational functions in x as coefficients. When y = 0 he found explicitly that P satisfies the hypergeometric equation in this form: d2y (1 - z) d log z 2 dy , (A + Bz) d log Z + (A - B'z)y = O. Riemann could therefore connect his P-functions with the functions F (a,/3, y, x) of Gauss: F (a b c, z) = const.

The conclusions to be drawn from the above exercises are that u -- fo x dt ( 1 - - t 2 ) 1/2 does not define a single-valued function of x, since u(x) is only well-defined up to a multiple of 2rr, but that x = x(u) is a well defined function (the sine function), which moreover is periodic; sin(u) = sin(u + 2rrn), and satisfies an algebraic addition theorem. What about the similar integral V -- fo x dt (1 - - t 4 ) 1/2 ? Deduce from Euler's algebraic addition theorem that it is also periodic, with period 2w = 4 f l dt ( l -- t4) 1/2" Jo Whenever x is real and the path of integration is real, it defines a singlevalued function of v, as Legendre was clearly aware.

Papers, II, 88, 89]. Independently of Gauss, Kummer pointed out that several of these substitutions could be spotted without going through the argument above. ,',z) is obtained. Kummer's analysis presents a complete answer to the first problem: what are the allowable changes of variable for the general hypergeometric equation? It immediately raises the second question: what are the solutions themselves? As Kummer pointed out (w there are many relationships between the solutions. Some, in any case, are equal to others, for example ( x) F(a, fl, y , x ) = (1 - x)Y-'~-#F(y - a , y - f l , y , x ) = (1-x)-~F a,y-fl, y, x-l- In fact, the six families of four solutions rearrange themselves into six different families of four equal solutions thus: 1, 2, 17, and 18; 3, 4, 19 and 20; 5, 6, 21, 22; 7, 8, 23, 24; 9, 12, 13, 15; and 10, 11, 14, 16.

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