Liouville's Theorem (other)

Liouville's theorem has various meanings, all mathematical results named after

Joseph 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 Thérèse ...

: * In complex analysis, see

Liouville's theorem (complex analysis) In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function f for which there ...

** There is also a related theorem on harmonic functions * In conformal mappings, see

Liouville's theorem (conformal mappings) In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of R''n'', where ''n'' > 2, can be expre ...

* In Hamiltonian mechanics, see

Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the trajectorie ...

and

Liouville–Arnold theorem In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with ''n'' degrees of freedom, there are also ''n'' independent, Poisson commuting first integrals of motion, and the energy level set ...

* In linear differential equations, see

Liouville's formula In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the s ...

* In transcendence theory and

diophantine approximations In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by r ...

, the theorem that any Liouville number is transcendental * In differential algebra, see

Liouville's theorem (differential algebra) In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions ...

* In differential geometry, see

Liouville's equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gel ...

* In coarse-grained modelling, see

in coarse graining phase space in classical physics and fine graining of states in quantum physics (von Neumann density matrix) {{mathdab