Search results
Results From The WOW.Com Content Network
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.
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 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 mathematical physics, Liouville made two fundamental contributions: the Sturm–Liouville theory, which was joint work with Charles François Sturm, and is now a standard procedure to solve certain types of integral equations by developing into eigenfunctions, and the fact (also known as Liouville's theorem) that time evolution is measure ...
Liouville's theorem is sometimes presented as a theorem in differential Galois theory, but this is not strictly true. The theorem can be proved without any use of Galois theory . Furthermore, the Galois group of a simple antiderivative is either trivial (if no field extension is required to express it), or is simply the additive group of the ...
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 ...
As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere [d] is constant. Thus any non-constant entire function must have a singularity at the complex point at infinity , either a pole for a polynomial or an essential singularity for a transcendental entire function.