7-2-54. Felix Klein to H. Poincaré, English translation
Leipzig, 9 July 1881
By way of a quick reply to your letter, I have something like the following to say.
1. It is fine with me that you to have quoted that passage of my letter. Up to now I have only your first Note of 27 June.11endnote: 1 Poincaré (1881). As for the name you gave that class of functions, I was quite surprised; but then I myself had no more than noticed the existence of these groups.22endnote: 2 Before sending this letter to Poincaré, Klein showed it to Georges Brunel, who then cited this remark in his own letter to Poincaré of 14 July, 1881 (§ 4-15-3). For my part, I will use neither “Fuchsian” nor “Kleinian” but remain with my “functions invariant under linear transformations”.
2. What I said about the value of Riemann’s principles was not precise enough. There can be no doubt that Dirichlet’s principle must be abandoned as not at all conclusive. However, it can be completely replaced by more rigorous methods of proof. You will find this expounded in more detail in a work by Schwarz that I have just recently seen (in connection with my course) and in which you will find information on the determination of the constants, which was only indicated in Borchardt’s Journal (you must in any case examine the memoirs published in Volumes 70, 74, and 75 of Borchardt’s Journal); the work of Schwarz in question is in the Berliner Monatsberichten 1870, pp. 767–795.
3. The general existence proof I mentioned last time remains valid, naturally, for groups made up of arbitrary analytic (not necessarily linear) substitutions. It is remarkable that in this sense every group of operations defines functions remaining unchanged by them. “Discontinuous groups” have the advantage that they have associated single-valued functions, a very fundamental property, moreover. Might one be able to master higher cases by means of single-valued functions of several variables as was the custom in connection with the case treated by Riemann in § 12 relating to the Jacobi inversion problem?
Enough for today. In the meantime, with Mr. Brunel I have looked over my works, notably the lecture notes from 1877–78 and 78–79 (which I had reworked back then), and he will shortly write to you about these.
With the greatest respect, your devoted
Prof. Dr. F. Klein.
Time-stamp: "28.04.2021 16:52"
- Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92, pp. 1484–1487. Cited by: endnote 1.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. Cited by: 7-2-54. Felix Klein to H. Poincaré, English translation.