Carl Gustav Jacob Jacobi edit
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Carl Gustav Jacob Jacobi edit
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| Carl Jacobi | |
 Carl Gustav Jacob Jacobi
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| Born | December 10, 1804 Potsdam, Kingdom of Prussia |
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| Died | February 18, 1851 (aged 46) Berlin, Kingdom of Prussia |
| Residence |  Prussia |
| Nationality | Prussian |
| Fields | Mathematician |
| Institutions | Königsberg University |
| Alma mater | University of Berlin |
| Doctoral advisor | Enno Dirksen |
| Doctoral students | Paul Albert Gordan Otto Hesse |
| Known for | Jacobi's elliptic functions Jacobian Jacobi symbol Jacobi identity |
Carl Gustav Jacob Jacobi (December 10, 1804 – February 18, 1851) was a Prussian mathematician, widely considered to be the most inspiring teacher of his time1 and one of the greatest mathematicians of all time.2
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He was born of Jewish parentage in Potsdam. 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 mathematics at Königsberg University, and this chair he filled until 1842.
Jacobi suffered a breakdown from overwork in 1843. He then visited Italy for a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner until his death. During the Revolution of 1848 Jacobi was politically involved and unsuccessfully presented his parliamentary candidature on behalf of a Liberal club. This led, after the suppression of the revolution, to his royal grant being cut off – but his fame and reputation were such that it was soon resumed.
Jacobi's grave is preserved at a cemetery in the Kreuzberg section of Berlin, the Friedhof I der Dreifaltigkeits-Kirchengemeinde (61 Baruther Street). His grave is close to that of Johann Encke, the astronomer.
The crater Jacobi on the Moon is named after him.
Jacobi wrote the classic treatise (1829) on elliptic functions, of great importance in mathematical physics, because of the need to "integrate second order kinetic energy equations". The motion equations in rotational form are integrable only for the three cases of the pendulum, the symmetric top in a gravitational field, and a freely spinning body, wherein solutions are in terms of elliptic functions. See Jacobi's elliptic functions.
Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the 2 square and four-square theorems of Pierre de Fermat. He also proved similar results for 6 and 8 squares. The Jacobi theta functions, so frequently applied in the study of hypergeometric series, were named in his honor.
He proved the functional equation for the theta function.
He proved the Jacobi triple product formula and many other results in q-series.
He gave new proofs of quadratic reciprocity, made contributions to higher reciprocity laws, investigated continued fractions and invented Jacobi sums.
In 1841 he reintroduced the partial derivative ∂ notation of Legendre, which was to become standard.
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 (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 Alfred Clebsch (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 manuscripts, 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.
The phrase 'Invert, always invert,' ('man muss immer umkehren') is associated with Jacobi for he believed that it is in the nature of things that the solution of many hard problems can be clarified by re-expressing them in inverse form.
The solution of the Jacobi inversion problem for the hyperelliptic Abel map by Weierstrass in 1854 required the introduction of the hyperlliptic theta function and later the general Riemann theta function for algebraic curves of arbitrary genus. The complex torus associated to a genus g algebraic curve, obtained by quotienting 
by the lattice of periods is referred to as the Jacobian or the Jacobi variety. This method of inversion, and its subsequent extension by Weierstrass and Riemann to arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic integrals and the Jacobi, or Weierstrass elliptic functions.
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