Sur les Fonctions Abeliennes in American Journal of Mathematics, Volume VIII, 1886, pp. 289-342. Baltimore: Johns Hopkins University, 1886.

Baltimore: John Hopkins University, 1886. 1st Edition. FIRST EDITION OF POINCARE’S PROOF & A GENERALIZATION OF TWO THEOREMS OF KARL WEIERSTRASS, a German mathematician frequently cited as the ‘father of modern analysis.’ “Henri Poincaré (1854-1912) was a mathematician, theoretical physicist and a philosopher of science famous for discoveries in several fields and referred to as the last polymath, one who could make significant contributions in multiple areas of mathematics and the physical sciences” (Stanford Encyclopedia of Philosophy).

While the proof Poincare published here had appeared in a French journal, Poincare wanted it placed in American Journal of Mathematic, the journal offered here, so that he could both reproduce and expand upon it. As with many of Poincare’s work, this one exists, or involves, the interaction between various branches of mathematics.

A translation of the first paragraph of Poincare’s paper reads: ““I have given in the Bulletin de la Societe mathematique de France (t. 12, page 124) a proof and a generalization of two theorems of M. Weierstrass. I wish to reproduce them here succinctly by making some additions which are essential to the proof” (Poincare, 289). The paper proceeds under six headings, respectively: Reduction of Integrals; Singular Case of Reduction; Generalization of the Theorem of Abel; Intermediary Functions; Transformation; Sum of Zeros” (Poincare, 289).

Poincare’s proof and generalization relates to Weierstrass’s work on Abelian functions and algebraic geometry. In fact, “as soon as he came into contact with the work of Riemann and Weierstrass on Abelian functions and algebraic geometry, Poincare was very much attracted by those fields. His papers on these subjects occupy in his complete works as much space as those on automorphic functions, their dated ranging from 1881 to 1911. One of the main ideas in these papers is that of "reduction" of Abelian functions. Generalizing particular cases studied by Jacobi, Weierstrass, and Picard, Poincare proved the general "complete reducibility" theorem... Abelian varieties can be decomposed in sums of "simple" abelian varieties having finite intersection. Poincare noted further that Abelian functions corresponding to reducible varieties (and even to products of elliptic curves, that is, Abelian varieties of dimension 1) are "dense" among all Abelian functions - a result that enabled him to extend and generalize many of Riemann's results on theta functions, and to investigate the special properties of the theta functions corresponding to the Jacobian varieties of algebraic curves. (Dictionary of Scientific Biography, Vol. 11 p. 54). Item #1389

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