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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 ex ...
** There is also a related theorem on harmonic functions * In conformal mappings, see Liouville's theorem (conformal mappings) * 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 trajectori ...
and Liouville–Arnold theorem * In linear differential equations, see Liouville's formula * In transcendence theory and diophantine approximations, 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 functio ...
* In differential geometry, see Liouville's equation * In coarse-grained modelling, see Liouville's equation in coarse graining phase space in classical physics and fine graining of states in quantum physics (von Neumann density matrix) {{mathdab