## 7-2-46. Felix Klein to H. Poincaré, English translation

Leipzig, 12 June [1881]

Address: Leipzig, Sophienstraße 10/II

Dear Sir,

Your three Notes in the Comptes rendus: ‘‘On fuchsian functions’’, which I first became acquainted with yesterday and then only fleetingly, are so closely related to the reflections and endeavors that have occupied me these last years, that I feel obliged to write to you.11endnote: 1 Poincaré 1881a; 1881b; 1881c; reed. Nörlund and Lebon (1916, 1–10). First of all I would like to tell you of various works of mine on elliptic functions published in Volumes XIV, XV, and XVII of the Mathematische Annalen. As far as modular elliptic functions are concerned, I dealt with only a special case of the independence relation that you consider; but a closer examination will show you that I did indeed have a general point of view. In this regard, I draw your attention to certain particular points:

Volume XIV, p. 128 deals with general functions representable by modular functions, independently of being connected with doubly periodic functions. Then follows, first in a special case, the important theorem on the fundamental polygon.

Volume XIV, pp. 159–160 is where I expound the result that every hypergeometric series can be represented by single-valued functions of suitable modular functions.

Volume XIV, p. 428ff. contains a table illustrating the mutual disposition of triangles with sides of circular arcs and angles $\frac{\pi}{7},\frac{\pi}{3},\frac{\pi}{2}$ (which is also an example from the classes of special functions studied by Halphen), apropos of which I should mention by the way that Mr. Schwarz has elaborated the case $\frac{\pi}{2},\frac{\pi}{4},\frac{\pi}{4}$ in Volume LXXV of Crelle’s Journal.22endnote: 2 Schwarz (1873).

Starting from p. 62 of Volume XVII, I present a rapid overview of the more mature conceptions of the theory of elliptic modular functions which in the meantime had occurred to me. I have not published anything on these; however I did present them during the summer of 1879 in a course at the Munich Polytechnic. My line of thought, which comes very close to your exposition at many points, was as follows:

1. Periodic and doubly periodic functions are just examples of single-valued functions admitting linear transformations. It is the task of modern analysis to determine all these functions.

2. The number of such transformations may be finite; these give the equations of the icosahedron, octahedron, …, that I investigated earlier (Math. Annalen IX, XII) and served as the point of departure for all of these sets of ideas.33endnote: 3 Klein (1876, 1877a, 1877b).

3. Groups of infinitely many linear transformations, giving rise to useful groups (discontinuous groups, in your terminology), are obtained for example when one starts with a polygon with circular arcs as sides, such that these circles cut a fixed circle orthogonally and have angles equal to exact simple fractions of $\pi$.

4. One should investigate all such functions (as indeed you have already begun doing); however, in order to attain concrete goals, let us confine ourselves to triangles of circular arcs, and, in particular, to elliptic modular functions.

Since then I have been much occupied with these questions, also in discussions with other mathematicians, but except to say that I have not yet obtained any definitive result, this is really not the place for them. I would like to limit myself to what I have published or lectured on. Perhaps I should have made contact with you sooner, or with one of your friends, such as Mr. Picard. (When the occasion presents itself, would you draw Mr. Picard’s attention to Annalen, XIV, p. 122, § 8?) For, the line of development of the set of ideas that have engaged you for the last 2 or 3 years, is, in actuality, very close to mine. I would also be very happy if this first letter led to a continuing correspondence. It is true that at the moment other business takes me from these questions, but I am the more encouraged to take them up again in that next Winter I am to give a course on differential equations.

Please convey my compliments to Mr. Hermite. I have often thought of starting a correspondence with him, and would have done so long ago — doubtless to my great profit — if it were not for the language problem. As you may perhaps know, I was in Paris long enough to learn to speak and write in French; however, in the meantime this ability has faded through disuse.

With the greatest respect,

Prof. Dr. F. Klein

PTrL. Translated by Robert Burns from the original German in Nörlund (1923, 97–99) (§ 4-47-1), and published in Saint-Gervais (2016).

Time-stamp: "20.07.2021 13:18"

## References

• F. Klein (1876) Ueber binäre Formen mit linearen Transformationen in sich selbst. Mathematische Annalen 9, pp. 183–208. Cited by: endnote 3.
• F. Klein (1877a) Ueber lineare Differentialgleichungen. Mathematische Annalen 12, pp. 167–179. Cited by: endnote 3.
• F. Klein (1877b) Weitere Untersuchungen über das Ikosaeder. Mathematische Annalen 12, pp. 503–560. Cited by: endnote 3.
• N. E. Nörlund and E. Lebon (Eds.) (1916) Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. Cited by: endnote 1.
• N. E. Nörlund (1923) Correspondance d’Henri Poincaré et de Felix Klein. Acta mathematica 39, pp. 94–132.
• H. Poincaré (1881a) Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92 (7), pp. 333–335. Cited by: endnote 1.
• H. Poincaré (1881b) Sur les fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92 (8), pp. 395–398. Cited by: endnote 1.
• H. Poincaré (1881c) Sur une nouvelle application et quelques applications importantes des fonctions fuchsiennes. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris 92, pp. 859–861. Cited by: endnote 1.
• H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich.
• H. A. Schwarz (1873) Ueber diejenigen Fälle, in welchen die Gaussiche hypergeometrische Reihe eine algebraische Function ihres vierten Elements darstellt. Journal für die reine und angewandte Mathematik 75 (4), pp. 292–335. Cited by: endnote 2.