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    On Riemann's Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises
    On Riemann's Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises
    On Riemann's Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises
    Ebook147 pages3 hours

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

    By Felix Klein

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    LanguageEnglish
    PublisherDover Publications
    Release dateOct 17, 2018
    ISBN9780486835105
    On Riemann's Theory of Algebraic Functions and Their Integrals: A Supplement to the Usual Treatises
    Author

    Felix Klein

    Dr. Felix Klein ist ein deutscher Jurist und Diplomat. Er ist auf Völkerrecht spezialisiert und seit 2018 Beauftragter der Bundesregierung für jüdisches Leben in Deutschland und den Kampf gegen Antisemitismus.

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      On Riemann's Theory of Algebraic Functions and Their Integrals - Felix Klein

      Conclusion

      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 me a 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 plane, 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 Göttingen, anyone who has followed Riemann’s speculations as they have come down to us, partly in fragments†, 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.

      BORKUM,

      Oct. 7, 1881.

      * Theory of Functions treated geometrically. Part 1, Winter-semester 1880—81, Part II, Summer-semester 1881.

      † I denote thus the contents of the investigations with which Riemann was concerned in the first part of his Theory of the Abelian Functions. The theory of the θ-functions, as developed in the second part of the same treatise, is in the first place, as we know, of an essentially different character, and is excluded from the following presentation as it was from my course of lectures.

      * Cf. C. Neumann, Math. Ann., t. x., pp. 569—571. Kirchhoff, Berl. Monatsber., 1875, pp. 487—497. Töpler, Pogg. Ann., t. CLX., pp. 375—388.

      † Ges. Werke, pp. 494 et seq.

      * Compare in particular the investigations on this subject by C. Neumann and Schwarz. The general case of closed surfaces (which is the most important for us in what follows) is indeed, as yet, nowhere explicitly and completely dealt with. Herr Schwarz contents himself with a few indications with respect to these surfaces (Berl. Monatsber., 1870, pp. 767 et seq.) and Herr C. Neumann only considers those cases in which functions are to be determined by means of known values on the boundary.

      PART I.

      INTRODUCTORY REMARKS.

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

      The physical interpretation of those functions of x + iy which are dealt with in the following pages is well known*. The principles on which it is based are here indicated, solely for completeness.

      Let w=u + iv, z=x + iy,w = f(z). Then we have, primarily,

      In these equations we

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