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He was born of Jewish parentage in 1804. He studied at Berlin University, where he obtained the degree of Doctor of Philosophy in 1825, his thesis being an analytical discussion of the theory of fractions. In 1827 he became extraordinary and in 1829 ordinary professor of mathematicsMathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of "figures and numbers". In the formalist view, it is the investigation of axiomatically defined abstract structures at Königsberg, and this chair he filled until 1842Events February 21 John J. Greenough patents the sewing machine. March 5 Over 500 Mexican troops led by Rafael Vasquez invade Texas briefly occupy San Antonio and then head back to the Rio Grande. This is the first such invasion since the Texas Revolution. Jacobi suffered a breakdown from overwork in 1843Events February 6 The first minstrel show in the United States The Virginia Minstrels opens (Bowery Amphitheatre in New York City). February 11 Giuseppe Verdi's opera I Lombardi premieres in Milan May 18 The Disruption of the Church of Scotland took place. He then visited ItalyThe Italian Republic or Italy ( Italian: Italia is a country in the south of Europe, consisting mainly of a boot-shaped peninsula together with two large islands in the Mediterranean Sea: Sicily and Sardinia. To the north, where it borders France, Switzer for a few months to regain his health. On his return he removed to Berlin, where he lived as a royal pensioner till his death.
Jacobi wrote the classic treatise ( 1829) on elliptic functionIn complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. The elliptic functions can be seen as analogs of the trigonometric functions (which have a single period only).s, of great importance in mathematical physics, because of the need to "integrate second order kinetic energy equations". The motion equation s in rotational form are integrable only for the three cases of the pendulumA gravity pendulum is a weight on the end of a rigid rod, which, when given some initial lift from the vertical position, will swing back and forth under the influence of gravity over its central (lowest) point. A torsion pendulum consists of a body suspe, the symmetric top in a gravitational field, and a freely spinning body, wherein solutions are in terms of elliptic functions.
Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the polygonal number theorem of Pierre de Fermat. The Jacobi theta functions, so frequently applied in the study of hypergeometric series, were named in his honor.
His investigations in elliptic functions, the theory of which he established upon quite a new basis, and more particularly his development of the theta function, 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 n2 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.
In his 1835 paper, Jacobi proved the following:
Jacobi reduced the general quintic equation to the form,
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. While contributing to celestial mechanics, Jacobi ( 1836) introduced the Jacobi integral for a sidereal coordinate system .
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 . Perhaps his most publicized work is Hamilton-Jacobi theory in rational mechanics.
Students of vector theory often encounter the Jacobi identity, those studying differential equations often encounter the Jacobian determinant, and those working in number theory and cryptography use the Jacobi symbol.