Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    Only $11.99/month after trial. Cancel anytime.

    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

    ()

    Read preview
    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.

    Related to On Riemann's Theory of Algebraic Functions and Their Integrals

    Titles in the series (100)

    View More
    View More

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • Article

      Physicists Attack Math’s $1,000,000 Question

      Apr 4, 2017

      Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system.

      4 min read

    • Article

      Math’s Beautiful Monsters: How a destructive idea paved the way for modern math.

      Apr 3, 2014

      Much like its creator, Karl Weierstrass’ monster came from nowhere. After four years at university spent drinking and fencing, Weierstrass had left empty handed. He eventually took a teaching course and spent most of the 1850s as a schoolteacher in B

      10 min read

    • Article

      Why Mathematicians Should Stop Naming Things After Each Other

      Sep 2, 2020

      Any student of modern math must know what it feels like to drown in a well of telescoping terminology. For a high-profile example, let’s take the Calabi-Yau manifold, made famous by string theory. A Calabi-Yau manifold is a compact, complex Kähler ma

      6 min read

    • Article

      How Math’s Most Famous Proof Nearly Broke: Andrew Wiles thought he had a solution to an age-old puzzle. Until it began to unravel.

      May 28, 2015

      Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into t

      8 min read

    • Article

      The Math That Takes Newton Into the Quantum World

      Feb 28, 2019

      In my 50s, too old to become a real expert, I have finally fallen in love with algebraic geometry. As the name suggests, this is the study of geometry using algebra. Around 1637, René Descartes laid the groundwork for this subject by taking a plane,

      9 min read

    • Article

      Math’s Beautiful Monsters

      Oct 26, 2017

      Much like its creator, Karl Weierstrass’ monster came from nowhere. After four years at university spent drinking and fencing, Weierstrass had left empty handed. He eventually took a teaching course and spent most of the 1850s as a schoolteacher in B

      10 min read

    • Article

      Math After COVID-19

      Apr 28, 2020

      Math After COVID-19
      4 min read

    • Article

      Kolmogorov Complexity and Our Search for Meaning

      Aug 2, 2018

      Was it a chance encounter when you met that special someone or was there some deeper reason for it? What about that strange dream last night—was that just the random ramblings of the synapses of your brain or did it reveal something deep about your u

      8 min read

    • Article

      The Aesthetic Beauty of Math

      Jul 22, 2019

      The Aesthetic Beauty of Math
      5 min read

    • Article

      The Deepest Uncertainty: When a hypothesis is neither true nor false.

      Jun 6, 2013

      Georg Cantor died in 1918 in a sanatorium in Halle, Germany. A pre-eminent mathematician, he had laid the foundation for the theory of infinite numbers in the 1870s. At the time, his ideas received hostile opposition from prominent mathematicians in

      4 min read

    • Article

      Euclid as Founding Father: We hold this mathematics to be self evident.

      Oct 13, 2016

      Despite the man’s awkward gestures, unkempt hair, and ill-fitting suit, it was one of the most extraordinary speeches that Reverend John Gulliver had ever heard. It was March 1860, and the venue was Norwich, Connecticut. The following morning Gullive

      8 min read

    • Article

      Actually, There Is a Time Like the Present

      Nov 30, 2017

      We’ve all heard it before: There’s no time like the present. Broadly speaking, of course, it means to “seize the opportunity right now,” or maybe in my case, to avoid procrastinating. From a psychological perspective, this makes a lot of sense. As hu

      4 min read

    • Article

      U.S. Mathematician Becomes First Woman To Win Abel Prize, 'Math's Nobel'

      Mar 19, 2019

      "I find that I am bored with anything I understand," Karen Uhlenbeck once said. That sentiment is part of why she won what many call the Nobel of mathematics Tuesday.

      3 min read

    • Article

      What Keeps Black Holes From Expanding Everywhere?

      Dec 10, 2018

      Black holes are somehow able to grow constantly without changing their size. Physics might finally be able to explain why.

      4 min read

    • Article

      How Math’s Most Famous Proof Nearly Broke

      Dec 20, 2018

      Andrew Wiles gave a series of lectures cryptically titled “Modular Forms, Elliptic Curves, and Galois Representations” at a mathematics conference in Cambridge, England, in June 0f 1993. His argument was long and technical. Finally, 20 minutes into t

      8 min read

    • Article

      The Universe Began With a Big Melt, Not a Big Bang

      Oct 5, 2017

      There are two tantalizing mysteries about our universe, one dealing with its final fate and the other with its beginning, that have intrigued cosmologists for decades. The community has always believed these to be independent problems—but what if the

      10 min read

    • Article

      How Physics Is Like Three-Chord Rock: Like a set of common chords, the same math appears in diverse fields of physics.

      Jun 12, 2014

      A few years back, 12 million of us clicked over to watch the “Pachelbel Rant” on YouTube. You might remember it. Strumming repetitive chords on his guitar, comedian Rob Paravonian confessed that when he was a cellist, he couldn’t stand the Pachelbel

      6 min read

    • Article

      The Scientific Problem That Must Be Experienced: To understand turbulence we need the intuitive perspective of art.

      Jul 3, 2014

      When the German physicist Arnold Sommerfeld assigned his most brilliant student a subject for his doctoral thesis in 1923, he admitted that “I would not have proposed a topic of this difficulty to any of my other pupils.” Those others included such g

      16 min read

    Related categories

    Reviews for On Riemann's Theory of Algebraic Functions and Their Integrals

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      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

      Enjoying the preview?
      Page 1 of 1