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ON RIEMANNS THEORY OF ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS: A Supplement to the Usual Treatises
by Felix Klein
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| Editorial Review |
| Editorial Review |
| German mathematician FELIX KLEIN (1849-1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. In his scholarly supplement to Riemann's complex mathematical theory, rather than offer proofs in support of the theorem, Klein chose to offer this exposition and annotation, first published in 1893, in an effort to broaden and deepen understanding. This approach makes Klein's commentary an essential element of any mathematics scholar's library. |
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| Product Details |
- Publisher: Cosimo Classics
- ISBN-10: 1-60206-327-3
- ISBN-13: 978-1-60206-327-3
- Amazon.com Sales Rank #2006169
- Published on: April 15, 2007
- Number of items: 1
- Binding: Paperback
- 92 pages
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| Customer Review |
Viktor Blasjo: Riemann surfaces through fluid dynamics  |
| 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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