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In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge c {\displaystyle c} of its Virasoro symmetry algebra , but it is unitary only if
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 equation appears in the study of isothermal coordinates in differential geometry: the independent variables x,y are the coordinates, while f can be described as the conformal factor with respect to the flat metric. Occasionally it is the square f 2 that is referred to as the conformal factor, instead of f itself.
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol ... Liouville's constant [33] 0.11000 10000 ...
One of these examples is Liouville's constant L = 0.110001000000000000000001 … , {\displaystyle L=0.110001000000000000000001\ldots ,} in which the n th digit after the decimal point is 1 if n {\displaystyle n} is the factorial of a positive integer and 0 otherwise.
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844 [1]), states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a positive number M {\displaystyle M} such that | f ( z ) | ≤ M ...
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
Therefore, g has to be constant on I, because otherwise we would obtain a contradiction to the mean value theorem (applied separately to the real and imaginary part in the complex-valued case). Since g(x 0) = det Φ(x 0), Liouville's formula follows by solving the definition of g for det Φ(x).