On Riemann's Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises

by Felix Klein

Overview

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.

Product Details

ISBN-13:	9781602063273
Publisher:	Cosimo Classics
Publication date:	04/15/2007
Pages:	92
Product dimensions:	5.50(w) x 8.50(h) x 0.22(d)

About the Author

One of the greatest German mathematicians of his era, Felix Klein (1849–1925) taught at the University of Göttingen from 1887 until his retirement in 1913. He made major contributions to many areas of mathematics, including group theory, complex analysis, and non-Euclidean geometry. Dover also publishes Klein's classic two-volume work, Elementary Mathematics from an Advanced Standpoint, part one of which focuses on geometry and part two on arithmetic, algebra, and analysis.

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PREFACE

THE pamphlet which I here lay before the public, has grown from lectures delivered during the past year, in which, among other objects, I had in view a presentation of Riemann s theory of algebraic functions and their integrals. Lectures on higher mathematics offer peculiar difficulties; with the best will of the lecturer they ultimately fulfil a very modest purpose. Being usually intended to give a systematic development of the subject, they are either confined to the elements or are lost amid details. I thought it well in this case, as previously in others, to adopt the opposite course. I assumed that the ordinary presentation, as given in text-books on the elements of Riemann's theory, was known ; moreover, when particular points required to be more fully dealt with, I referred to the fundamental monographs. But to compensate for this, I devoted great care to the presentation of the true train of thought, and endeavoured to obtain a general view of the scope and efficiency of the methods. I believe I have frequently obtained good results by these means, though, of course, only with a gifted audience; experience will show whether this pamphlet, based on the same principles, will prove equally useful.

A presentation of the kind attempted is necessarily very subjective, and the more so in the case of Riemann's theory, since but scanty material for the purpose is to be found explicitly given in Riemann's papers. I am not sure that I should ever have reached a well-defined conception of the whole subject, had not Herr Prym, many years ago (1874), in the course of an opportune conversation, made mea communication which has increased in importance to me the longer I have thought over the matter. He told me that Riemann s surfaces originally are not necessarily many-sheeted surfaces over the planei but that, on the contrary, complex functions of position can be studied on arbitrarily given curved surfaces in exactly the same way as on the surfaces over the plane. The following presentation will sufficiently show how valuable this remark has been to me. In natural combination with this there are certain physical considerations which have been lately developed, although restricted to simpler cases, from various points of view. I have not hesitated to take these physical conceptions as the starting-point of my presentation. Riemann, as we know, used Dirichlet's Principle in their place in his writings. But I have no doubt that he started from precisely those physical problems, and then, in order to give what was physically evident the support of mathematical reasoning, he afterwards substituted Dirichlet's Principle. Anyone who clearly understands the conditions under which Riemann worked in Gottingen, anyone who has followed Riemann's speculations as they have come down to us, partly in fragments fy will, I think, share my opinion. — However that may be, the physical method seemed the true one for my purpose. For it is well known that Dirichlet's Principle is not sufficient for the actual foundation of the theorems to be established; moreover, the heuristic element, which to me was all-important, is brought out far more prominently by the physical method. Hence the constant introduction of intuitive considerations, where a proof by analysis would not have been difficult and might have been simpler, hence also the repeated illustration of general results by examples and figures.

In this connection I must not omit to mention an important restriction to which I have adhered in the following pages. We all know the circuitous and difficult considerations by which, of late years, part at least of those theorems of Riemann which are here dealt with have been proved in a reliable manner. These considerations are entirely neglected in what follows and I thus forego the use of any except intuitive bases for the theorems to be enunciated. In fact such proofs must in no way be mixed up with the sequence of thought I have attempted to preserve; otherwise the result is a presentation unsatisfactory from all points of view. But they should assuredly follow after, and I hope, when opportunity offers, to complete in this sense the present pamphlet.

For the rest, the scope and limits of my presentation speak for themselves. The frequent use of my friends' publications and of my own on kindred subjects had a secondary purpose important to me for personal reasons: I wished to give my audience a guide, to help them to find for themselves the reciprocal connections among these papers, and their position with respect to the general conception put forth in these pages. As for the new problems which offer themselves in great number, I have only allowed myself to investigate them as far as seemed consistent with the general aim of this pamphlet. Nevertheless I should like to draw attention to the theorems on the conformal representation of arbitrary surfaces which I have worked out in the last Part; I followed these out the more readily that Riemann makes a remarkable statement about this subject at the end of his Dissertation.

One more remark in conclusion to obviate a misunderstanding which might otherwise arise from the foregoing words. Although I have attempted, in the case of algebraic functions and their integrals, to follow the original chain of ideas which I assumed to be Riemann's, I by no means include the whole of what he intended in the theory of functions. The said functions were for him an example only, in the treatment of which, it is true, he was particularly fortunate. Inasmuch as he wished to include all possible functions of complex variables, he had in mind far more general methods of determination than those we employ in the following pages; methods of determination in which physical analogy, here deemed a sufficient basis, fails us. Compare, in this connection, § 19 of his Dissertation, compare also his work on the hypergeometrical series. — With reference to this, I must explain that I have no wish to draw aside from these more general considerations by giving a presentation of a special part, complete in itself. My innermost conviction rather is that they are destined to play, in the developments of the modern Theory of Functions, an important and prominent part.

(Continues…)

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Table of Contents

Part I Introductory Remarks.

1 Steady Streamings in the Plane as an Interpretation of the Functions of x + iy 1

2 Consideration of the Infinities of w = f(z) 5

3 Rational Functions and their Integrals. Derivation of the Infinities of higher Order from those of lower Order 9

4 Experimental Production of these Streamings 12

5 Transition to the Surface of a Sphere. Streamings on arbitrary curved Surfaces 15

6 Connection between the foregoing Theory and the Functions of a complex Argument 19

7 Streamings on the Sphere resumed. Riemann's general Problem 21

Part II Riemann's Theory.

8 Classification of closed Surfaces according to the Value of the Integer p 23

9 Preliminary Determination of steady Streamings on arbitrary Surfaces 26

10 The most general steady Streaming. Proof of the Impossibility of other Streamings 29

11 Illustration of the Streamings by means of Examples 32

12 On the Composition of the most general Function of Position from single Summands 37

13 On the Multiformity of the Functions. Special Treatment of multiform Functions 40

14 The ordinary Riemann's Surfaces over the x + iyPlane 43

15 The Anchor-ring, p = 1, and the two=sheeted surface over the plane with four Branch-points 46

16 Functions of x + iy which correspond to the streamings already investigated 51

17 Scope and Significance of the previous Investigations 55

18 Extension of the Theory 56

Part III Conclusions.

19 On the Moduli of Algebraical Equations 59

20 Conformal Representation of closed Surfaces upon themselves 64

21 Special Treatment of symmetrical Surfaces 66

22 Conformal Representation of different closed Surfaces upon each other 70

23 Surfaces with Boundaries and unifacial Surfaces 72

24 Conclusion 75