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Liouville's theorem: Every holomorphic function: ... There is a short proof of the fundamental theorem of algebra using Liouville's theorem. [3] Proof ...
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.
Thus, on an intuitive level, the theorem states that the only elementary antiderivatives are the "simple" functions plus a finite number of logarithms of "simple" functions. A proof of Liouville's theorem can be found in section 12.4 of Geddes, et al. [4] See Lützen's scientific bibliography for a sketch of Liouville's original proof [5 ...
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
A Liouville number is irrational but does not have this property, so it cannot be algebraic and must be transcendental. The following lemma is usually known as Liouville's theorem (on diophantine approximation), there being several results known as Liouville's theorem.
The theorem was proven in its original form by Liouville in 1853 for functions on with canonical symplectic structure. It was generalized to the setting of symplectic manifolds by Arnold, who gave a proof in his textbook Mathematical Methods of Classical Mechanics published 1974.
Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below.
There is a powerful theorem that two complex numbers that are algebraically dependent belong to the same Mahler class. [24] [31] This allows construction of new transcendental numbers, such as the sum of a Liouville number with e or π.