Monday, October 31, 2011

Karl Gustav Jacob Jacobi

Dec 10 1804 - Feb 18 1851
Born Potsdam, Germany. Died Berlin, Germany.

Karl Gustav Jacob Jacobi founded the theory of elliptic functions.

Jacobi's father was a banker and his family were prosperous so he received a good education at the University of Berlin. He obtained his Ph.D. in 1825 and taught mathematics at the University of Königsberg from 1826 until his death, being appointed to a chair in 1832.

He founded the theory of elliptic functions based on four theta functions. His "Fundamenta nova theoria functionum ellipticarum" in 1829 and its later supplements made basic contributions to the theory of elliptic functions.

In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is imaginary. This result prompted much further work in this area, in particular by Liouville and Cauchy.

Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics.

He also worked on determinants and studied the functional determinant now called the Jacobian. Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy. However Jacobi wrote a long memoir "De determinantibus functionalibus" in 1841 devoted to the this determinant. He proves, among many other things, that if a set of n functions in n variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero.

Jacobi's reputation as an excellent teacher attracted many students. He introduced the seminar method to teach students the latest advances in mathematics.

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