黎曼代数函数及积分On Riemannis theory of algebraic functions and their integrals
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作 者：Felix
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ISBN：9780486495521
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简介
A great researcher, writer, and teacher in an era of tremendous math-ematical 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 comprehen-sively, and this book--a consideration of the investigations in the first part of Riemann's Theory of Abelian Functions--is a prime exam-ple of his expository powers.
After introducing Riemann's approach to multiple-value functions and the geometrical representation of these functions by what later became known as Riemann surfaces, Klein concentrates on the kinds of functions that can be defined on these surfaces, confining the treatment to rational functions and their integrals. He demonstrates how Riemann's mathematical ideas about Abelian integrals can be arrived at by thinking in terms of the flow of electric current on sur-faces, an excellent example of how contemplation of physical matters can suggest mathematical themes.
Klein's primary concern is preserving the sequence of thought and offering intuitive explanations of Riemann's notions, rather than fur-nishing detailed proofs. In addition to its significant contribution in the area of complex functions, this volume's formulation of the topo-
logical equivalents of Riemann's surfaces is extremely useful and remains one of the best introductions to the origins of topological problems.
目录
PART I.INTRODUCTORY REMARKS.
SECT.
1.Steady Streamings in the Plane as an Interpretation of the Functions of x + iy
2.Consideration of the Infinities of w=f(z)
3.Rational Functions and their Integrals.Derivation of the Infinities of higher Order from those of lower Order
4.Experimental Production of these Streamings
5.Transition to the Surface of a Sphere.Streamings on arbitrary curved Surfaces
6.Connection between the foregoing Theory and the Functions of a complex Argument
7.Streamings on the Sphere resumed.Riemann's general Problem
PART II. RIEMANN'S THEORY
8.Classification of closed Surfaces according to the Value of the Integer p
9.Preliminary Determination of steady Streamings on arbitrary Surikces
10.The most general steady Streaming.Proof of the Impossi- bility of other Streamings
11.Illustration of the Streamings by means of Examples
12.On the Composition of the most general Function of Position from single Summands
13.On the Multiformity of the Functions.Special Treatment of multiform Functions
14.The ordinary Riemann's Surfaces over the x+iy Plane
15.The Anchor-ring, n=1, and the two-sheeted Surface over the Plane with four Branch-points.
16.Functions of x+iy which correspond to the Streamings already investigated
17.Scope and Significance of the previous Investigations
18.Extension of the Theory
PART III.CONCLUSIONS.
19.On the Moduli of Algebraical Equations
20.Conformal Representation of closed Surfaces upon themselves
21.Special Treatment of symmetrical Surfaces
22.Conformal Representation of different closed Surfaces upon each other
23.Surfaces with Boundaries and unifacial Surfaces
24.Conclusion
黎曼代数函数及积分On Riemannis theory of algebraic functions and their integrals
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