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In number theory, a Liouville number is a real number with the property that, for every ... The following lemma is usually known as Liouville's theorem ...
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 ...
In ergodic theory and dynamical systems, motivated by the physical considerations given so far, there is a corresponding result also referred to as Liouville's theorem.In Hamiltonian mechanics, the phase space is a smooth manifold that comes naturally equipped with a smooth measure (locally, this measure is the 6n-dimensional Lebesgue measure).
In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental In differential algebra, see Liouville's theorem (differential algebra) In differential geometry, see Liouville's equation
Transcendental number theory is a branch ... Roth's work effectively ended the work started by Liouville, and his theorem allowed ... A Liouville number is defined to ...
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 ...
which does not satisfy Liouville's theorem, whichever degree n is chosen. This link between Diophantine approximations and transcendental number theory continues to the present day. Many of the proof techniques are shared between the two areas.
Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology, but also mathematical physics and even astronomy. He is remembered particularly for Liouville's theorem in complex analysis.