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2020-12-18 20:04:44
ABEL, Niels Henrik
Mémoire sur une propriété générale d'une classe trés-étendue de fonctions transcendantes, présenté à l'Académie le 30 Octobre 1826, pp. 176-264 in: Mémoires présentés par divers savants a l'Académie Royale des Scines de l'Institute de France . Tome Septième
Imprimé par autorisation du roi a l'Imprimerie Royale, Paris, 1841. 'A MONUMENT MORE LASTING THAN BRONZE' (LEGENDRE). First edition, journal issue, of "the most important single result in the theory of integrals of algebraic functions . Abel's theorem" (Bottazzini & Gray, p. 236), a result which Legendre called 'a monument more lasting than bronze'. It can be regarded as the birth of algebraic geometry. Abel had already begun to radically transform and generalize the theory of elliptic functions when, in July 1826, he visited Paris, hoping to make the acquaintance of the great mathematicians there. "The visit to Paris was to prove disappointing. The university vacations had just begun when Abel arrived, he found that they were aloof and difficult to approach; it was only in passing that he met Legendre, whose main interest in his old age was elliptic integrals, Abel's own specialty. For presentation to the French Academy of Sciences Abel had reserved a paper that he considered his masterpiece. It dealt with the sum of integrals of a given algebraic function. Abel's theorem states that any such sum can be expressed as a fixed number p of these integrals, with integration arguments that are algebraic functions of the original arguments. The minimal number p is the genus of the algebraic function, and this is the first occurrence of this fundamental quantity. Abel's theorem is a vast generalization of Euler's relation for elliptic integrals. Abel spent his first months in Paris completing his great memoir; it is one of his longest papers and includes a broad theory with applic … [Click Below for Full Description]
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