His investigations in elliptic functions, the theory of which he established upon quite a new basis, and more particularly his development of the thetafunction[?], as given in his great treatise Fundamenta nova theoriae functionum ellipticarum (Königsberg, 1829), and in later papers in Crelle's Journal[?], constitute his grandest analytical discoveries. Second in importance only to these are his researches in differential equations, notably the theory of the last multiplier[?], which is fully treated in his Vorlesungen über Dynamik, edited by R. F. A. Clebsch (Berlin, 1866). It was in analytical development that Jacobi’s peculiar power mainly lay, and he made many important contributions of this kind to other departments of mathematics, as a glance at the long list of papers that were published by him in Crelle’s Journal and elsewhere from 1826 onwards will sufficiently indicate. He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n^{2} differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations. Valuable also are his papers on Abelian transcendents[?], and his investigations in the theory of numbers, in which latter department he mainly supplements the labours of K. F. Gauss. The planetary theory and other particular dynamical problems likewise occupied his attention from time to time. He left a vast store of manuscript, portions of which have been published at intervals in Crelle's Journal. His other works include Comnienlatio de transformatione integralis duplicis indefiniti in formam simpliciorem (1832), Canon arithmeticus (1839), and Opuscula mathematica (1846—1857). His Gesammelte Werke (1881–1891) were published by the Berlin Academy[?].
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