Liouville, Joseph

▪ French mathematician

born March 24, 1809, Saint-Omer, France

died September 8, 1882, Paris

      French mathematician known for his work in analysis, differential geometry, and number theory and for his discovery of transcendental numbers—i.e., numbers that are not the roots of algebraic equations having rational coefficients. He was also influential as a journal editor and teacher.

      Liouville, the son of an army captain, was educated in Paris at the École Polytechnique from 1825 to 1827 and then at the École Nationale des Ponts et Chaussées (“National School of Bridges and Roads”) until 1830. At the École Polytechnique, Liouville was taught by André-Marie Ampère (Ampère, André-Marie), who recognized his talent and encouraged him to follow his course on mathematical physics at the Collège de France. In 1836 Liouville founded and became editor of the Journal des Mathématiques Pures et Appliquées (“Journal of Pure and Applied Mathematics”), sometimes known as the Journal de Liouville, which did much to raise and maintain the standard of French mathematics throughout the 19th century. The manuscripts of the French mathematician Évariste Galois (Galois, Évariste) were first published by Liouville in 1846, 14 years after Galois's death.

      In 1833 Liouville was appointed professor at the École Centrale des Arts et Manufactures, and in 1838 he became professor of analysis and mechanics at the École Polytechnique, a position that he held until 1851, when he was elected a professor of mathematics at the Collège de France. In 1839 he was elected a member of the astronomy section of the French Academy of Sciences (Sciences, Academy of), and the following year he was elected a member of the prestigious Bureau of Longitudes.

      At the beginning of his career, Liouville worked on electrodynamics and the theory of heat. During the early 1830s he created the first comprehensive theory of fractional calculus, the theory that generalizes the meaning of differential and integral operators. This was followed by his theory of integration in finite terms (1832–33), the main goals of which were to decide whether given algebraic functions have integrals that can be expressed in finite (or elementary) terms. He also worked in differential equations (differential equation) and boundary value problems, and, together with Charles-François Sturm (Sturm, Charles-François)—the two were devoted friends—he published a series of articles (1836–37) that created a completely new subject in mathematical analysis. Sturm-Liouville theory (Sturm-Liouville problem), which underwent substantial generalization and rigorization in the late 19th century, became of major importance in 20th-century mathematical physics as well as in the theory of integral equations (integral equation). In 1844 Liouville was the first to prove the existence of transcendental numbers, and he constructed an infinite class of such numbers. Liouville's theorem, concerning the measure-preserving property of Hamiltonian dynamics (Hamiltonian function) (conservation of total energy), is now known to be basic to statistical mechanics and measure theory.

      In analysis Liouville was the first to deduce the theory of doubly periodic functions (functions with two distinct periods whose ratio is not a real number) from general theorems (including his own) in the theory of analytic functions of a complex variable (complex number) (also known as holomorphic functions or regular functions; a complex-valued function defined and differentiable over some subset of the complex number plane). In number theory he produced more than 200 publications, most of which are in the form of short notes. Although nearly all of this work was published without indication of the means by which he had obtained his results, proofs have since been provided. Altogether, Liouville's publications comprise about 400 memoirs, articles, and notes.

Additional Reading

Jesper Lützen, Joseph Liouville, 1809–1882: Master of Pure and Applied Mathematics (1990), reviews Liouville's life and epoch in the first part and delves deeper into his work in the second half.

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