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Carl Jacobi: Leader in mathematical analysis
CARL GUSTAV Jacobi was born on December 10, 1804 (Potsdam, Germany), the son of a prosperous banker. He received education at home: his first teacher was one of his maternal uncles who prepared Jacobi to enter the local gymnasium in his twelfth year. He graduated in 1821 and entered University of Berlin, specialising in philogy and mathematics.
Like Gauss, he settled for the latter. The teacher allowed Jacobi to work on his own, as he rebelled at learning mathematics by rote.
In August 1825, Jacobi received the Ph.D. degree for a thesis on partial fractions. After working as a lecturer for six months at the University of Berlin, he moved in 1826 to Koenisberg. He returned in 1843 as professor to Berlin and served till the end of his life (February 18, 1851).
From the very beginning of his career, it was evident that he was a born teacher. Later, when he began developing his own ideas at an amazing speed, he established the reputation as the most inspiring teacher of his time.
Jacobi adopted a unique style to train students in research by taking up his own discoveries in the class and letting the students see the creation of a new subject taking place before them.
For it was his firm conviction that many students endlessly work till they have mastered what has been done by others.
Few could ever acquire the capability for independent work, as they put off attempting on their own initiative.
To drive home his point to a talented but hesitant young man who lacked the drive to pursue Jacobi used the following simile: ``Your father would never have married, and you would not be here now, if he had insisted on knowing all the girls in the world before marrying one'' (Men of Mathematics by E.T. Bell, Simon & Schuster, New York, 1965).
Jacobi's entire career was spent in teaching and research except one ghastly interlude in politics and occasional trips to Italy to recuperate his health.
family fortune collapsed in 1840. Gauss, (1777-1855) whose work overlapped many of Jacobi's discoveries feared some disastrous effect on Jacobi's mathematical output. But Bessel (1784-1813) assured him: `such a talent cannot be destroyed'. .
On the advice of an imbecile physician, Jacobi began entering in politics. This debacle ruined him: he was 45 and had his wife and seven small children to support.
A friend took care of the family, while Jacobi took up residence in a dingy hotel and continued his researches.
The most celebrated results of Jacobi were those in elliptic functions (1829) which won him praise from the legendary mathematician Lagrange, (1786-1813) who had spent some 40 years in the study of elliptic integrals.
Jacobi broke new ground in the theory of numbers, by using elliptic functions to prove Fermat's assertion: any integer is the sum of the squares of no more than four integers.
In 1841 he published a classic paper where he established the theory of determinants and its basic properties.
The results of Jacobi were in some respects parallel to the work of his rival Henrik Abel (1802-29), Jacobi made his great start in entire ignorance of his rival's contribution.
The private papers of Gauss show that he had anticipated both Jacobi and Abel by as much as 27 years in some of their discoveries.
What Gauss, Abel and Jacobi saw, was by inverting the functional relationship u=f (v), one obtains a more useful function v=f (u).
The most striking property of these transcendental functions was they have a double periodicity, that is v=f (u)=f (u+m) = f (u+n) where m and n are complex numbers.
Whereas trignometic functions have a real period (2 pi ) and the function e{+i}{+x} has an imaginary period {lcub}2 (pi) i{rcub}, the elliptic functions have double periodicity.
So impressed was Jacobi with the simplicity achieved through simple inversion of the functional relationship in elliptic integrals that he gave, for secret of success in maths, the advice ``you must always invert.''
R. Parthasarathy
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