Carl Jacobi – Wikipedia

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from Wikipedia, L’Encilopedia Libera.

Carl Gustav Jacob Jacobi
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Carl Gustav Jacob Jacobi (Potsdam, 10 December 1804 – Berlin, 18 February 1851) was a German mathematician and teacher.

He was born from a Jewish family in 1804. He studied at the University of Berlin, where he obtained the doctoral title in 1825, with a dissertation containing an analytical discussion of the theory of fractions. In 1827 he became extraordinary professor and in 1829 full professor of mathematics in Königsberg (the current Kaliningrad), and retained this chair until 1842. Jacobi suffered from a physical shoulder strap caused by excessive work in 1843 and moved to Italy for a few months to regain health. On his return he moved to Berlin, where he lived until his death, due to a smallpox infection.

Jacobi in 1829 wrote his classic treatise on the elliptical functions in which he faced the problem of “integrating the second -order equations obtained from kinetic energy”. Some cases in which these motorcycle equations can be integrated, with solutions expressed in terms of elliptical functions, are the pendulum , the symmetrical (or Lagrange) top in a constant gravitational field and the motion of a rigid body suspended for the center of gravity (or Euler’s top).

Jacobi was also the first mathematician to apply the elliptical functions to the theory of numbers, in particular showing the theorem of the polygonal number of Pierre de Ferrat. The Theta Jacobian function, applied in the studies of the hypergeometric series, was so called in his honor.

His research on elliptical functions, the theory of which he founded on an innovative basis, and more in particular the development of the Theta function, as described in his treaty The foundation of the new theory of the functions of the elliptical (Königsberg, 1829), and in subsequent articles on the Journal Für Die Reine und Angewandte Mathematik by August Crelle, constitute his major analytical discoveries. Other important research concerned the differential equations, in particular the theory of the last multiplier, which is treated in his work Lectures on dynamics , published by Alfred Clebsch (Berlin, 1866).

It was in the analytical developments that emerge more evidently the skills of Jacobi; He provided important contributions of this kind to other areas of mathematics, as is evident from the articles he published in the Journal of Crelle and elsewhere in the years from 1826 onwards. He was one of the first lovers of the determinants theory; in particular, he invented the functional determinant formed by n 2 differential coefficients of n -Uple of functions in n Independent variables, which today bears its name (Jacobiano) and who has played an important role in many analysis searches.

In an article from 1835 Jacobi showed that:

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If a univariate (univocal) function is periodic, then it cannot have more than two periods and the relationship between these periods cannot be a real number.

By studying the General Quintica Jacobi equation, he manages to reduce it to the form

He also wrote articles on Abelian transcendants and his research in numbers, within which he took care above all to complete Gauss’s work.

Planetary theory and other particular problems of similar dynamics committed its attention from time to time. While providing contributions to celestial mechanics, Jacobi in 1836 introduced the integral of Jacobi relating to a celestial coordinate system.

He left a large amount of manuscripts, part of which was published several times in the Journal de Crelle. His other works include The commentary on the transformation of the integral double in the form of simplification (1832), Canon arithmeticus (1839) and Brochures mathematics (1846-1857). His Collected Works (1881 – 1891) was published by the Berlin Academy. Another important work is made up of Hamilton-Jacobi theory in the context of rational mechanics.

The result of his works allowed the development in different fields: the identity of Jacobi applied to the theory of vectors; The determinant Jacobiano in the field of differential equations and the symbol of Jacobi applied to the theory of numbers and to encryption.

A crater on the moon was dedicated to him, the Jacobi crater and an asteroid, 12040 Jacobi.

  • Eric Temple Bell (1937): Men of Mathematics , Simon and Schuster, New York
  • David Hestes (1986): New Foundations of Classical Mechanics , Kluwer Adademic Publishers, Dordrecht

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