By Jeremy Gray

This booklet is a research of ways a specific imaginative and prescient of the team spirit of arithmetic, referred to as geometric functionality conception, used to be created within the nineteenth century. The primary concentration is at the convergence of 3 mathematical subject matters: the hypergeometric and similar linear differential equations, team thought, and non-Euclidean geometry.

The textual content for this re-creation has been significantly increased and revised, and the present appendices enriched with ancient money owed of the Riemann--Hilbert challenge, the uniformization theorem, Picard-Vessiot idea, and the hypergeometric equation in better dimensions. The routines were retained, making it attainable to take advantage of the publication as a significant other to arithmetic classes on the graduate point.

This paintings remains to be the one up to date scholarly account of the heritage of a department of arithmetic that maintains to generate very important examine, for which the maths has been the party for one of the most profound paintings via a number of nineteenth century figures: Riemann, Fuchs, Dedekind, Klein, and Poincar.

Reviews of the 1st version:

"The volume...is one amongst the main attention-grabbing books at the historical past of arithmetic released in contemporary times.... very stimulating examining for either historians of recent arithmetic and mathematicians as well." ---Mathematical reports

"The ebook includes an grand wealth of fabric on the subject of the algebra, geometry, and research of the 19th century.... Written with actual old standpoint and transparent exposition, this publication is really challenging to place down." ---Zentralblatt fur Mathematik

"...must studying for each critical pupil of 19th century mathematics...represents a considerable contribution towards filling what's often stated to be an important hole within the ancient literature."---ISIS

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