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 theorem on harmonic functions In conformal mappings, see Liouville's theorem (conformal mappings) In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) 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 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)