## 7-2-70. 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*.^{1}^{1}endnote:
^{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.^{2}^{2}endnote:
^{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

- Œuvres d’Henri Poincaré, Volume 2. Gauthier-Villars, Paris. link1 Cited by: endnote 2.
- Mémoire sur les fonctions fuchsiennes. Acta mathematica 1, pp. 193–294. link1 Cited by: endnote 2.
- Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires. Mathematische Annalen 19, pp. 553–564. link1 Cited by: endnote 1.
- Théorie des groupes fuchsiens. Acta mathematica 1, pp. 1–62. link1 Cited by: endnote 2.
- Mémoire sur les groupes kleinéens. Acta mathematica 3, pp. 49–92. link1 Cited by: endnote 2.
- Mémoire sur les fonctions zétafuchsiennes. Acta mathematica 5, pp. 209–278. link1 Cited by: endnote 2.
- Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich. link1, link2 Cited by: 7-2-70. Felix Klein to H. Poincaré, English translation.