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In number theory, a Liouville number is a real number with the property that, for every positive integer, there exists a pair of integers (,) with > such that < | | <. The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations.
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 ...
The irrationality exponent or Liouville–Roth irrationality measure is given by setting (,) =, [1] a definition adapting the one of Liouville numbers — the irrationality exponent () is defined for real numbers to be the supremum of the set of such that < | | < is satisfied by an infinite number of coprime integer pairs (,) with >.
In conformal mappings, see Liouville's theorem (conformal mappings) In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) and Liouville–Arnold theorem; In linear differential equations, see Liouville's formula; In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental
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 ...
Ax–Kochen theorem (number theory) Baker's theorem (number theory) Barban–Davenport–Halberstam theorem (analytic number theory) Basel problem (mathematical analysis) Beatty's theorem (Diophantine approximation) Behrend's theorem (number theory) Bertrand's postulate (number theory) Birch's theorem (algebraic number theory) Bombieri's ...
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