Liouville's theorem

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).
*In conformal mappings, see Liouville's theorem (conformal mappings).
*In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian).
*In number theory, the theorem that any Liouville number is transcendental, "or" the lemma involved on diophantine approximation.
*In differential algebra, a theorem which allows one to determine when a given elementary function has no elementary integral, see Liouville's theorem (differential algebra)

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  • Liouville's theorem (complex analysis) — In complex analysis, Liouville s theorem, named after Joseph Liouville, states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that | f ( z )| ≤ M for all… …   Wikipedia

  • 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 trajectories… …   Wikipedia

  • Liouville's theorem (conformal mappings) — In mathematics, Liouville s theorem is a theorem about conformal mappings in Euclidean space. It states that any conformal mapping on a domain of R n , where n > 2, can be expressed as a composition of translations, similarities, orthogonal… …   Wikipedia

  • liouville's theorem — (ˈ)lyü|vēlz noun Usage: usually capitalized L Etymology: after Joseph Liouville died 1882 French mathematician : a theorem in fluid dynamics: the density of any selected part of a stream of fluid that does no work and that has no work done on it… …   Useful english dictionary

  • Liouville's theorem — Math. the theorem that every function of a complex variable, bounded and differentiable for all finite values of the variable, is a constant function. [named after J. LIOUVILLE] * * * …   Universalium

  • Liouville number — In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that A Liouville number can thus be approximated quite closely by a sequence of… …   Wikipedia

  • Liouville, Joseph — ▪ French mathematician born March 24, 1809, Saint Omer, France died September 8, 1882, Paris       French mathematician known for his work in analysis, differential geometry, and number theory and for his discovery of transcendental numbers i.e …   Universalium

  • Liouville's equation — For Liouville s equation in dynamical systems, see Liouville s theorem (Hamiltonian). In differential geometry, Liouville s equation, named after Joseph Liouville, is the equation satisfied by the conformal factor f of a metric f^2 (dx^2 + dy^2)… …   Wikipedia

  • Liouville-Gleichung —   [lju vil ; nach J. Liouville], partielle Differenzialgleichung der statistischen Mechanik für die Verteilungsfunktion f(N) (r1,. .., rN; pN …   Universal-Lexikon

  • Liouville-Theorem — Der Satz von Liouville (auch Liouville Theorem genannt, nach Joseph Liouville) ist eine direkte Folge aus der Liouville Gleichung und besagt, dass das von benachbarten Trajektorien im Phasenraum eingeschlossene (mehrdimensionale) Volumen konstant …   Deutsch Wikipedia

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