7-2-55. Felix Klein to H. Poincaré, English translation

Leipzig, 4 December 1881

Sophienstraße 10/II

Dear Sir!

After having long reflected only in passing on the questions of mutual interest to us, this morning I took the opportunity of reading together the different communications that you have published successively in the Comptes rendus. I see that you have now definitely proved (as of 8 August): ‘‘que toutes les équations différentielles linéaires à coefficients algébriques s’intègre par les fonctions zétafuchsiennes’’ and ‘‘que les coordonnées des points d’une courbe algébrique quelconque s’expriment par des fonctions fuchsiennes d’une variable auxiliaire’’.11endnote: 1 Poincaré (1881), reed. Nörlund and Lebon (1916, 29–31). “[T]hat every linear differential equation with algebraic coefficients is integrable by means of zetafuchsian functions and that the coordinates of any algebraic curve whatever can be expressed via fuchsian functions of an auxiliary variable.” While congratulating you on the results you have obtained, I would also like to put a proposition to you respecting both your interests and mine equally. I would ask you to to send me, for Mathematische Annalen, a more or less long article, or, if you don’t have the time to write up such a work, then a letter, expounding, in broad strokes, your points of view and results. I would then write an accompanying note in which I would describe how I view the whole matter and how at this juncture the research program that you are now pursuing has served as a fundamental guiding principle for my work on modular functions. Of course, I would submit this note to you for your approval prior to publication. Such a publication would have a double effect: first, it would definitely draw the attention of the readers of Math. Annalen to your work, undoubtedly a desirable outcome for you; and second, your work would be presented to a large general readership, at the same time demonstrating the connections with my work as they actually are. As I know from what you wrote to me, you intend to analyse these equations in a detailed memoir; but writing it will take time, and I would like an announcement to be made in the Annalen also.

For my part, I have meanwhile written up a little treatise on ‘‘Riemann’s theory’’ which may be of interest to you since it presents a version of the concept of a Riemann surface that I believe R. himself actually worked on.22endnote: 2 Klein (1882), reed. Klein (1923, 499–573). Perhaps Mr. Brunel has told you of this. I have also been busy lately with different existence proofs designed to replace Dirichlet’s principle, and I am convinced that the methods expounded by Schwarz in the Berliner Monatsberichten, 1870, p. 767 et seqq. in any case suffice completely to yield, for example, the general theorem I wrote to you about once or twice this past summer.33endnote: 3 Schwarz (1870).

With the greatest respect,

F. Klein

PTrL. Translated by R. Burns from the original German (§ 4-47-10), in Saint-Gervais (2016), with slight revisions by S.A. Walter. See also the French translation (§ 7-2-36).

Time-stamp: "28.04.2021 17:33"

Notes

  • 1 Poincaré (1881), reed. Nörlund and Lebon (1916, 29–31). “[T]hat every linear differential equation with algebraic coefficients is integrable by means of zetafuchsian functions and that the coordinates of any algebraic curve whatever can be expressed via fuchsian functions of an auxiliary variable.”
  • 2 Klein (1882), reed. Klein (1923, 499–573).
  • 3 Schwarz (1870).

References

  • F. Klein (1882) Ueber Riemann’s Theorie der algebraischen Functionen und ihrer Integrale. Teubner, Leipzig. Link Cited by: endnote 2.
  • F. Klein (1923) Gesammelte mathematische Abhandlungen, Volume 3. Springer, Berlin. Link Cited by: endnote 2.
  • N. E. Nörlund and E. Lebon (Eds.) (1916) Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. Link Cited by: endnote 1.
  • H. Poincaré (1881) Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 93, pp. 301–303. Link Cited by: endnote 1.
  • H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. Link Cited by: 7-2-55. Felix Klein to H. Poincaré, English translation.
  • H. A. Schwarz (1870) Ueber die Integration der partiellen Differentialgleichung 2ux22uy2=0\frac{\partial^{2}u}{\partial x^{2}}{\partial^{2}u}{\partial y^{2}}=0 unter vorgeschriebenen Grenz- und Unstetigkeitsbedingungen. Monatsberichte der königliche Akademie der Wissenschaften zu Berlin, pp. 767–795. Link Cited by: endnote 3.