Poincare, H [Jules Henri]Biblio.com
Sur les Fonctions Abeliennes in American Journal of Mathematics Volume VIII 1886 pp. 289-342. Baltimore: Johns Hopkins University 1886
18861389Baltimore: 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. <br /> <br /> 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. <br /> <br /> 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. <br /> <br /> 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. CONDITION & DETAILS: Full volume handsomely bound in half red leather and marbled boards scuffed and rubbed at the edges and spine; raised bands at the spine as well as gilt-lettering. Ex-libris bookplate front paste-down library "Due Date" label tipped-in rfep small library number sticker spine. No other library markings. 4to. Clean and bright throughout. Very good. John Hopkins University hardcover
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