Joseph Liouville
Joseph Liouville | |
|---|---|
Liouville in 1868 | |
| Born | 24 March 1809 |
| Died | 8 September 1882 (aged 73) |
| Alma mater | École Polytechnique |
| Known for | Sturm–Liouville theory Liouville's equation Liouville's theorem (complex analysis) Liouville number Liouville-Green method Liouville's theorem (Hamiltonian) Liouville-Arnold theorem Liouville field theory Liouville's theorem (differential algebra) Liouville function Liouville's formula |
| Scientific career | |
| Fields | Mathematics |
| Institutions | École Centrale Paris École Polytechnique |
| Thesis | Sur le développement des fonctions ou parties de fonctions en séries de sinus et de cosinus, dont on fait usage dans un grand nombre de questions de Mécanique et de Physique (1836) |
| Doctoral advisor | Siméon Poisson Louis Jacques Thénard |
| Doctoral students | Eugène Charles Catalan Nikolai Bugaev |
Joseph Liouville (/ˌliːuˈvɪl/ LEE-oo-VIL; French: [ʒozɛf ljuvil]; 24 March 1809 – 8 September 1882)[1][2] was a French mathematician who worked on a number of different fields in mathematics, including number theory, complex analysis, and mathematical physics.
The crater Liouville on the Moon is named after him.
Life and work
He was born in Saint-Omer in France on 24 March 1809.[3][4] His parents were Claude-Joseph Liouville (an army officer) and Thérèse Liouville (née Balland).
Liouville gained admission to the École Polytechnique in 1825 and graduated in 1827. Just like Augustin-Louis Cauchy before him, Liouville studied engineering at École des Ponts et Chaussées after graduating from the Polytechnique, but opted instead for a career in mathematics. After some years as an assistant at various institutions including the École Centrale Paris, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France in 1850 and a chair in mechanics at the Faculté des Sciences in 1857.
Besides his academic achievements, he was very talented in organisational matters. Liouville founded the Journal de Mathématiques Pures et Appliquées (which retains its high reputation up to today) in order to promote other mathematicians' work. He was the first to read, and to recognize the importance of, the unpublished work of Évariste Galois which appeared in his journal in 1846. Liouville was also involved in politics for some time, and he became a member of the Constituting Assembly in 1848. However, after his defeat in the legislative elections in 1849, he turned away from politics.
In 1851, he was elected a foreign member of the Royal Swedish Academy of Sciences. In 1853, he was elected as a member of the American Philosophical Society.[5]
Contributions
Analysis, algebra, and number theory
He demonstrated Liouville's theorem in complex analysis. He established the existence of non-elementary integrals and a criterion for integration in finite terms, that is, in terms of elementary functions.
In algebra, Liouville was one of the first to grasp the significance of the contributions of the late Évariste Galois, work had been forwarded to him by Auguste Chevalier, a friend of Galois.[6]
In number theory, he was the first to prove the existence of transcendental numbers; he used a construction with continued fractions (Liouville numbers).[7] The Liouville function, an important concept in number theory, is named in his honor. In his work on elliptic integrals, he demonstrated the transcendence of Abelian functions.
Mathematical physics
In mathematical physics, Liouville joinly developed the Sturm–Liouville theory with his friend, Jacques Charles François Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions. Their work was inspired by the analysis of heat diffusion in a cylinder by Jean-Baptiste Joseph Fourier.[8]
Liouville also the fact that the time evolution in phase space of a mechanical system was measure preserving, a result now known as Liouville's theorem in Hamiltonian mechanics. In the same context, Liouville introduced the notion of action-angle coordinates as a description of completely integrable systems. The modern formulation of this is sometimes called the Liouville–Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability.
In his study of electrodynamics, Liouville developed the Riemann-Liouville integral to consider differentiation and integration of a fractional order.[9]
See also
Notes
- ^ His death is registered the 9th of Septembre Etat civil de la ville de Paris, 6ème arrondissement.
- ^ Figaro du 10 décembre 1882
- ^ Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 2016-03-04. Retrieved 2017-04-28.
- ^ "Joseph Liouville | French mathematician | Britannica". www.britannica.com. Retrieved 2021-12-11.
- ^ "APS Member History". search.amphilsoc.org. Retrieved 2021-04-16.
- ^ Ehrhardt, Caroline (August 2011). "A quarrel between Joseph Liouville and Guillaume Libri at the French Academy of Sciences in the middle of the nineteenth century". Historia Mathematica. 38 (3): 389–414. doi:10.1016/j.hm.2011.02.002.
- ^ Joseph Liouville (May 1844). "Mémoires et communications". Comptes rendus de l'Académie des Sciences (in French). 18 (20, 21): 883–885, 910–911.
- ^ Grattan-Guinness, Ivor (May 1973). "More Recent Mathematics: Mathematical Thought from Ancient to Modern Times, Morris Kline". Science. 180 (4086): 627–8. doi:10.1126/science.180.4086.627.
- ^ Lützen, Jesper (1985). "Liouville's differential calculus of arbitrary order and its electrodynamical origin". Proceedings of the 19th Nordic Congress Mathematicians. Icelandic Mathematical Society: 149–160.
References
- O'Connor, John J.; Robertson, Edmund F., "Joseph Liouville", MacTutor History of Mathematics Archive, University of St Andrews
- Lützen, Jesper (1990), Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics, Studies in the History of Mathematics and Physical Sciences, vol. 15, Springer-Verlag, ISBN 3-540-97180-7
Further reading
- Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, vol. 76, Cambridge: Cambridge University Press, ISBN 978-0-521-17562-3, Zbl 1227.11002