Joseph Liouville ( ; ; 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.
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. 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. As a mathematician, he maintained contact with many foreign colleagues, including William Thomson (Lord Kelvin), Carl Gustav Jacob Jacobi, and Peter Gustav Lejeune Dirichlet.
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. 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.
Research on the solutions of algebraic equations spurred interest in algebraic and transcendental irrational numbers. 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.
Mathematical physics
Since the middle eighteenth century, mathematicians and physicists had been studying a variety of partial differential equations with boundary values using the separation of variables to resolve them into systems ordinary differential equations, which carried their own parameters. Solutions found for specific values of these parameters, called eigenvalues, were known as eigenfunctions. The separation of variables in different coordinate systems led to new special functions, such as the Bessel functions and Legendre polynomials, as eigenfunctions of ordinary differential equations. Liouville and his friend, Jacques Charles François Sturm, sought to tackle the general problem for any linear differential equations of the second order. In a series papers published in the 1830s, the two men established the Sturm–Liouville theory. it is now a standard procedure to solve certain types of integral equations. and may be viewed as a generalization of Fourier analysis. Liouville sought approximate solutions to linear second-order differential equations with spatially varying coefficients and obtained, in modern language, an asymptotic series. The Liouville–Green method was rediscovered in 1923 by Harold Jeffreys, and again in 1926 by Gregor Wentzel, Hans Kramers, Léon Brillouin, who were studying the Schrödinger equation of quantum mechanics. that phase space volume of a conservative mechanical system constant, a result now known as Liouville's theorem in Hamiltonian mechanics. In a related 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.
He also studied potential theory