title: Liouville's theorem
text: # Liouville's theorem
  
  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)
  