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This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of f {\displaystyle f} is not dense, then there is a complex number w {\displaystyle w} and a real number r > 0 {\displaystyle r>0} such that the open disk centered at w {\displaystyle w} with radius r {\displaystyle r} has ...
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 of the system—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time.
Joseph Liouville FRS FRSE FAS (/ ˌ l iː u ˈ v ɪ l / LEE-oo-VIL, French: [ʒozɛf ljuvil]; 24 March 1809 – 8 September 1882) [1] [2] was a French mathematician and engineer. Life and work [ edit ]
In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, [1] is a rigidity theorem about conformal mappings in Euclidean space.It states that every smooth conformal mapping on a domain of R n, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions).
In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, [1] [2] [3] places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions.
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
[2] This is the original form of Liouville's theorem and can be derived from it. [3] A holomorphic elliptic function is bounded since it takes on all of its values on the fundamental domain which is compact. So it is constant by Liouville's 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 level sets of all first integrals are compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only ...