On Riemann’s Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises (Dover Books on Mathematics) by Felix Klein (PDF)

Description

A great researcher, writer, and teacher in an era of tremendous mathematical ferment, Felix Klein (1849–1925) occupies a prominent place in the history of mathematics. His many talents included an ability to express complicated mathematical ideas directly and comprehensively, and this book, a consideration of the investigations in the first part of Riemann’s Theory of Abelian Functions, is a prime example of his expository powers.The treatment introduces Riemann’s approach to multiple-value functions and the geometrical representation of these functions by what later became known as Riemann surfaces. It further concentrates on the kinds of functions that can be defined on these surfaces, confining the treatment to rational functions and their integrals. The text then demonstrates how Riemann’s mathematical ideas about Abelian integrals can be arrived at by thinking in terms of the flow of electric current on surfaces. Klein’s primary concern is preserving the sequence of thought and offering intuitive explanations of Riemann’s notions, rather than furnishing detailed proofs. Deeply significant in the area of complex functions, this work constitutes one of the best introductions to the origins of topological problems.

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⭐Where a complex function is analytic it satisfies the Cauchy-Riemann equations, which implies that its real and imaginary part satisfy the Laplace equation, which means that the function corresponds to a steady fluid flow. Restricting ourselves to algebraic functions and their integrals, we can understand the corresponding flows by studying the infinities of these functions, and we can imagine electromagnetic set-ups that would produce such flows. “Now it seems possible, ab initio, to reverse the whole order of this discussion; to study the [flows] in the first place and thence to work out the theory of certain analytical functions” (p. 22). So we study this type of flows on general spheres with handles, and then show that this corresponds precisely to the algebraic functions and their integrals on their Riemann surfaces. In this way, without any of the usual analytic machinery, we reveal the qualitative insights of Riemann’s theory. In particular, “among the numerous accidental properties of the functions, we distinguish certain essential ones” (p. 56), namely the infinities, the Riemann surface, and the topological types of the cross-cuts. This concludes the study of Riemann’s theory proper, but for the last part of the book “we may bring forward the geometrical side of the subject and consider Riemann’s theory as a means of making the theory of conformal representation of one closed surface upon another accessible to analytical treatment” (p. 56).

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