POINCARÉ, HENRI.

Mémoire sur les Fonctions fuchsiennes.

[Berlin, Stockholm, Paris, F. & G. Beijer, 1882]. Large4to. As extracted from "Acta Mathematica", In "Acta Mathematica", volume 1. Clean and fine. Pp. 193-294.

First printing of Poincaré's famous paper which conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. It also constitute the second paper in Poincaré's exceedingly important series of six paper's which together represent the discovery of Automorphic Functions. "Before he was thirty years of age, Poincaré became world famous with his epoch-making discovery of the "automorphic functions" of one complex variable (or, as he called them, the "fuchsian" and "kleinean" functions)." (DSB).

These manuscripts, written between 28 June and 20 December 1880, show in detail how Poincaré exploited a series of insights to arrive at his first major contribution to mathematics: the discovery of the automorphic functions. In particular, the manuscripts corroborate Poincaré's introspective account of this discovery (1908), in which the real key to his discovery is given to be the recognition that the transformations he had used to define Fuchsian functions are identical with those of non-Euclidean geometry.

The idea was to come in an indirect way from the work of his doctoral thesis on differential equations. His results applied only to restricted classes of functions and Poincaré wanted to generalize these results but, as a route towards this, he looked for a class functions where solutions did not exist. This led him to functions he named Fuchsian functions after Lazarus Fuchs but were later named automorphic functions.

First editions and first publications of these epochmaking papers representing the discovery of "automorphic functions", or as Poincaré himself called them, the "Fuchsian" and "Kleinian" functions.
"By 1884 Poincaré published five major papers on automorphic functions in the first five volumes of the new Acta Mathematica. When the first of these was published in the first volume of the new Acta Mathematica, Kronecker warned the editor, Mittag-Leffler, that this immature and obscure article would kill the journal. Guided by the theory of elliptic functions, Poincarë invented a new class of automorphic functions. This class was obtained by considering the inverse function of the ratio of two linear independent solutions of an equation. Thus this entire class of linear diffrential equations is solved by the use of these new transcendental functions of Poincaré." (Morris Kline).

Poincaré explains how he discovered the Automorphic Functions: "For fifteen days I strove to prove that there could not be any functions like those I have since called Fuchsian functions, I was then very ignorant; every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds; I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a Class of Fuchsian functions, those which come from hypergeometric series; i had only to write out the results, which took but a few hours...the transformations that I had used to define the Fuchsian functions were identical with those of Non-Euclidean geometry..."

Order-nr.: 45910