Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work

He was born in
Saint-Omer
Saint-Omer (; ; Picard: ''Saint-Onmé'') is a commune and sub-prefecture of the Pas-de-Calais department in France.
It is west-northwest of Lille on the railway to Calais, and is located in the Artois province. The town is named after Sa ...
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
(, ; also known as Polytechnique or l'X ) is a ''grande école'' located in Palaiseau, France. It specializes in science and engineering and is a founding member of the Polytechnic Institute of Paris.
The school was founded in 1794 by mat ...
in 1825 and graduated in 1827. Just like
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
before him, Liouville studied engineering at
École des Ponts et Chaussées
École or Ecole may refer to:
* an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing in région Île-de-France
* Éco ...
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
École or Ecole may refer to:
* an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée)
* École (river), a tributary of the Seine flowing i ...
, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the
Collège de France
The (), formerly known as the or as the ''Collège impérial'' founded in 1530 by François I, is a higher education and research establishment () in France. It is located in Paris near La Sorbonne. The has been considered to be France's most ...
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
The ''Journal de Mathématiques Pures et Appliquées'' () is a French monthly scientific journal of mathematics, founded in 1836 by Joseph Liouville (editor: 1836–1874). The journal was originally published by Charles Louis Étienne Bachelier. ...
'' 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
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, ...
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.
Liouville worked in a number of different fields in mathematics, including
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
,
differential geometry and topology
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus, linear algebra and multil ...
, but also
mathematical physics
Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...
and even
astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
. He is remembered particularly for
Liouville's theorem in complex analysis. In number theory, he was the first to prove the existence of
transcendental number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
s by a construction using
continued fraction
A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...
s (
Liouville number
In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that
:0<\left, x-\frac\<\frac.
The inequality implies that Liouville numbers po ...
s). In mathematical physics, Liouville made two fundamental contributions: the
Sturm–Liouville theory
In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form
\frac \left (x) \frac\right+ q(x)y = -\lambda w(x) y
for given functions p(x), q(x) and w(x), together with some ...
, which was joint work with
Charles François Sturm
Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...
, and is now a standard procedure to solve certain types of
integral equation
In mathematical analysis, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,\ldots,x_n ; u(x_1,x_2 ...
s by developing into eigenfunctions, and the fact (also known as
Liouville's theorem) that time evolution is measure preserving for a
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
system. In Hamiltonian dynamics, Liouville also introduced the notion of
action-angle variables
In classical mechanics, action-angle variables are a set of canonical coordinates that are useful in characterizing the nature of commuting flows in integrable systems when the conserved energy level set is compact, and the commuting flows are co ...
as a description of completely
integrable systems
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. The modern formulation of this is sometimes called the Liouville–Arnold theorem, and the underlying concept of integrability is referred to as Liouville integrability. Additionally, Liouville developed the
Riemann-Liouville integral to consider differentiation and integration of a
fractional order.
In 1851, he was elected a foreign member of the
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...
. In 1853, he was elected as a member of the
American Philosophical Society
The American Philosophical Society (APS) is an American scholarly organization and learned society founded in 1743 in Philadelphia that promotes knowledge in the humanities and natural sciences through research, professional meetings, publicat ...
.
The crater
Liouville
Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
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 Thérès ...
on the
Moon
The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...
is named after him. So is the
Liouville function
The Liouville lambda function, denoted by and named after Joseph Liouville, is an important arithmetic function.
Its value is if is the product of an even number of prime numbers, and if it is the product of an odd number of primes.
Explicit ...
, an important function in number theory.
See also
*
List of things named after Joseph Liouville
*
Liouville's theorem (disambiguation) Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville:
* In complex analysis, see Liouville's theorem (complex analysis)
** There is also a related Harmonic functions#Liouville's theorem, theorem on harmon ...
Notes
References
*
*
* Lutzen J., "Liouville's differential calculus of arbitrary order and its electrodynamical origin", in ''Proc. 19th Nordic Congress Mathematicians''. 1985. Icelandic Mathematical Society, Reykjavik, pp. 149–160.
Further reading
*
External links
*
*
{{DEFAULTSORT:Liouville, Joseph
École Polytechnique alumni
École des Ponts ParisTech alumni
Corps des ponts
1809 births
1882 deaths
19th-century French mathematicians
French mathematical analysts
Members of the French Academy of Sciences
Members of the Royal Swedish Academy of Sciences
Foreign members of the Royal Society
Members of the Göttingen Academy of Sciences and Humanities
International members of the American Philosophical Society