7-2-69. H. Poincaré to Felix Klein, English translation

Paris 18 May 1882

Sir,

Your last letter interested me greatly, needless to say.

I now see clearly that your proof and mine differ at most in terminology and details. Consequently, it is probable that we don’t establish in the same way the analytic character of the relation between the two Mannigfaltigkeiten of which you speak. Myself, I connect this fact to the convergence of my series, but it is obvious that one can obtain the same result by other means.

Mr. Schwarz’s ideas have much greater scope; it is clear that the general theorem in question, if it were proven, would find application to the theory of a great many functions and in particular to that of functions defined by non-linear differential equations. It was in studying such equations that I came, for my part, to ask if a Riemann surface of infinitely many sheets could be extended over a disc, and in this regard I was led to the following problem, which would allow for a proof of the possibility of this extension:

We take a partial differential equation

 $X_{1}\frac{d^{2}u}{dx^{2}}+X_{2}\frac{d^{2}u}{dxdy}+X_{3}\frac{d^{2}u}{dy^{2}}% +X_{4}\frac{du}{dx}+X_{5}\frac{du}{dy}=0,$

and a half-circumference $AMBO$.

Here $X_{1}$, $X_{2}$, $X_{3}$, $X_{4}$, $X_{5}$ are given functions of $x$ and $y$; these functions are analytic inside the half-circumference, and cease to be so on its perimeter. Can we always find a function $\mu$ of $x$ and $y$ satisfying the equation, analytic inside the half-circumference, and tending to 1 as the point $x$, $y$ approaches the half-circumference and to 0 as it approaches the diameter $AOB$? All my efforts in this direction have proved fruitless up to now, but I hope that Mr. Schwarz, who has solved the problem in the simplest case, will be more fortunate than I.

I am sending you offprints of my earlier articles, and hope soon to be able to send you the other, more recent memoirs I mentioned, the offprints of which I should have in hand before long.

As for Mr. Hermite’s lithographed lectures, they are published by Hermann, Librairie des Lycées, rue de la Sorbonne; the subscription costs 12 Francs. I don’t believe the editor sends Mr. Hermite any offprints.

Please be assured of my most devoted sentiments and my sincere esteem.

Poincaré

PTrL. Translated by R. Burns in Saint-Gervais (2016) from the original French (§ 4-47-24), with minor revisions by S.A. Walter.

Time-stamp: "14.05.2021 18:11"

References

• H. P. d. Saint-Gervais (2016) Uniformization of Riemann Surfaces: Revisiting a Hundred-Year-Old Theorem. European Mathematical Society, Zurich.