## 7-2-68. Felix Klein to H. Poincaré, English translation

Leipzig, 19 Sept. 1882

Sophienstr. 10/II

Dear Sir!

While on my side I was about to finish a lengthy paper on the new functions, I reread once again your article in Volume 19 of Math. Annalen.11endnote: 1 Poincaré (1882b). There is a point I do not understand: in two places (in the middle of p. 558, and at the bottom of p. 560), you speak of “fonctions fuchsiennes” existing only in a space bounded by infinitely many circles all orthogonal to the fundamental circle. Now I know such functions very well (as I already wrote you three months ago), having as their natural boundary an infinitude of circles. However, the corresponding group always contains substitutions leaving invariant only a single limit circle, chosen arbitrarily. Now you define as “fuchsian” those functions all of whose substitutions are real (p. 554), and this definition remains essentially unchanged by the generalization on p. 557, where the real axis is replaced by an arbitrary circle. Thus the functions I am familiar with do not fall under your definition of “fuchsian”. Is this a misunderstanding on my part or an imprecision in the formulation on yours? As far as my own work is concerned, I confine myself to expounding the geometric viewpoint, by virtue of which I consider I have defined the new functions in Riemann’s sense. One finds, as is only natural, many points of contact with your geometric conception of the subject. The most general groups that I consider are generated by a certain number of “isolated” substitutions and a certain number of groups “with a fundamental circle” (which may be real or imaginary or reduced to a point) via “nesting in one another”. The theorems of my two Annalen notes then become special cases of the general theorem which may be expressed as follows: to every Riemann surface with arbitrarily prescribed branching and cuts there corresponds one and only one $\eta$-function of the type in question.

From Mittag-Leffler I hear that you too are now occupied with important work.22endnote: 2 See Poincaré (1882c), (1882a), (1883), (1884), and the reedition by Nörlund and Lebon (1916, 108–462). I need not tell you just how much I would be interested in knowing more about that work. If in a month’s time you’re in Paris, you will get to know my friend S. Lie, who has been visiting me for a few days and who, although not a function-theorist himself as yet, is very interested in the progress made of late in function theory.

With the greatest respect,

F. Klein

PTrL. Translated by R. Burns from the original German (§ 4-47-25) in Saint-Gervais (2016), with minor revisions by S.A. Walter. See also the French translation (§ 7-2-42), and Poincaré’s reply (§ 4-47-26).

Time-stamp: "14.05.2021 19:03"

## References

• N. E. Nörlund and E. Lebon (Eds.) (1916) Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. Cited by: endnote 2.
• H. Poincaré (1882a) Mémoire sur les fonctions fuchsiennes. Acta mathematica 1, pp. 193–294. Cited by: endnote 2.
• H. Poincaré (1882b) Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires. Mathematische Annalen 19, pp. 553–564. Cited by: endnote 1.
• H. Poincaré (1882c) Théorie des groupes fuchsiens. Acta mathematica 1, pp. 1–62. Cited by: endnote 2.
• H. Poincaré (1883) Mémoire sur les groupes kleinéens. Acta mathematica 3, pp. 49–92. Cited by: endnote 2.
• H. Poincaré (1884) Mémoire sur les fonctions zétafuchsiennes. Acta mathematica 5, pp. 209–278. Cited by: endnote 2.
• H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich.