Joseph Liouville ( LEE-oo-VIL; French: [ʒozɛf ljuvil]; 24 March 1809 – 8 September 1882) was a French mathematician who worked on a number of different fields in mathematics, including number theory, complex analysis, and mathematical physics. Many mathematical concepts are named after him, including Liouville's theorem in Hamiltonian mechanics and Liouville's theorem for complex numbers.
He was born in Saint-Omer in France on 24 March 1809. 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 began delivering lectures on mathematics at the Collège de France in 1851 secured a chair in rational mechanics at the Faculté des Sciences in 1857. However, he suffered under a heavy teaching load and his health started to deteriorate.
Liouville founded the Journal de Mathématiques Pures et Appliquées in 1836, basing it on Crelle's Journal. It soon became a leading mathematical periodical in France and became known as "Liouville's Journal" even after he had resigned as editor-in-chief in 1875. He was elected to the French Academy of Sciences in 1839, and became an associate member of the Bureau des Longitudes.
Liouville was also involved in politics for some time, and he became a member of the Constituting Assembly in following the 1848 Revolution. However, following the rise of Napoleon III to power, Liouville ended his political activities.
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. As a mathematician, he maintained contact with many foreign colleagues, including William Thomson (Lord Kelvin), Carl Gustav Jacob Jacobi, and Peter Gustav Lejeune Dirichlet. As a lecturer, he offered support and encouragement to many young talents, among them, Charles Hermite, Joseph Bertrand, and Joseph-Alfred Serret.
Analysis, algebra, and number theory
In 1835, Liouville established the existence of non-elementary integrals and a criterion for integration in finite terms, that is, in terms of elementary functions. He gave an example of such a non-elementary integral,
{\displaystyle \int {\frac {e^{x}}{x}}\,dx}
In 1838, Liouville published a method for establishing the existence of solutions to ordinary differential equations of the second order involving successive approximations, now attached to the name of Émile Picard, who gave a more general approach in the early 1890s.
In algebra, Liouville was one of the first to grasp the significance of the contributions of the late Évariste Galois, whose work had been forwarded to him by Auguste Chevalier, a friend of Galois. Liouville edited and published the work of Galois in his own journal in 1846, after which the Galois theory attracted the attention of many mathematicians, among them, Paolo Ruffini, Joseph-Alfred Serret, and Augustin-Louis Cauchy. By helping to popularize the works of Galois, Liouville participated, albeit indirectly, in the development of modern algebra in general and group theory in particular.
Research on the solutions of algebraic equations spurred interest in algebraic and transcendental irrational numbers. In 1844, Liouville was the first to prove the existence of transcendental numbers. He did so by demonstrating some results on approximating algebraic irrationals using rational numbers and established an inequality that served as a criterion for transcendence. In essence, his inequality proclaims that rationals are poor approximations of irrational algebraic numbers. He then gave an explicit example. He showed that any number of the form
{\displaystyle {\frac {a_{1}}{10}}+{\frac {a_{2}}{10^{2!}}}+{\frac {a_{3}}{10^{3!}}}+\cdots ,}
are integers from 0 to 9, is transcendental. Future demonstrations of the transcendence of Euler's number
(pi) by Ferdinand von Lindemann were based on Liouville's contributions.
The Liouville function, an important concept in number theory, is named in his honor.
In his work on elliptic integrals, he based the whole subject on the general properties of doubly periodic functions and he demonstrated the transcendence of Abelian functions. It was in this context that he discovered Liouville's theorem in complex analysis, bounded entire functions are constant. He also gave a short proof of the fundamental theorem of algebra.
A similar result is Liouville's theorem for harmonic functions, or solutions to Laplace's equation. It states that bounded harmonic functions in Euclidean space are constant. Edward Nelson gave a short proof in 1961, exploiting the mean-value property of harmonic functions.