JACOBI, KARL GUSTAV JACOB

born Dec. 10, 1804, Potsdam, Prussia died Feb. 18, 1851, Berlin German mathematician who, with Niels Henrik Abel of Norway, founded the theory of elliptic functions. In 1827 Jacobi became extraordinary professor and in 1829 ordinary professor of mathematics at the University of Knigsberg. He first became known through his work on number theory, which gained the admiration of Carl Gauss, the greatest mathematician of his day. Unaware of similar endeavours by Abel, Jacobi formulated a theory of elliptic functions based on four theta functions. The quotients of the theta functions yield the three Jacobian elliptic functions: sn z, cn z, and dn z. His results in elliptic functions were published in Fundamenta Nova Theoriae Functionum Ellipticarum (1829; New Foundations of the Theory of Elliptic Functions). In 1832 he demonstrated that just as elliptic functions can be obtained by inverting elliptic integrals, hyperelliptic functions can be obtained by inverting hyperelliptic integrals. This success led him to the formation of the theory of Abelian functions of p variables (where p is greater than or equal to 2). Jacobi's De Formatione et Proprietatibus Determinantium (1841; Concerning the Structure and Properties of Determinants) made pioneering contributions to the theory of determinants. He invented the functional determinant (formed from the n2 differential coefficients of n given functions with n independent variables) that bears his name and has played an important part in many analytical investigations. Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics. His Vorlesungen ber Dynamik (1866; Lectures on Dynamics) relates his work with differential equations and dynamics. The Hamilton-Jacobi equation plays a significant role in the presentation of quantum mechanics. He also made important studies of Abelian transcendents and the applications of elliptic functions to the theory of numbers.