Liouville showed that all Liouville numbers are transcendental.

Kurt Mahler showed in 1953 that π is also not a Liouville number.

For example, Liouville showed that g ( 4 ) is at most 53.

Any Liouville number must have unbounded partial quotients in its continued fraction expansion.

Using the explicit continued fraction expansion of e, one can show that e is not a Liouville number ( although the partial quotients in its continued fraction expansion are unbounded ).

In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exist integers p and q with q > 1 and such that

* In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental.

The Liouville function, denoted by λ ( n ) and named after Joseph Liouville, is an important function in number theory.

This allowed Liouville, in 1844 to produce the first explicit transcendental number.

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.

In number theory, he was the first to prove the existence of transcendental numbers by a construction using continued fractions ( Liouville numbers ).

So is the Liouville function, an important function in number theory.

* Joseph Liouville finds the first transcendental number

Three subfields of the complex numbers C have been suggested as encoding the notion of a " closed-form number "; in increasing order of generality, these are the EL numbers, Liouville numbers, and elementary numbers.

# REDIRECT Liouville number

# REDIRECT Liouville number

Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of,,, and.

The volume is said to be computed by the Liouville measure.

The volume of the energy shell, computed using the Liouville measure, is preserved under evolution.

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

The Liouville measure restricted to the energy surface Ω is the basis for the averages computed in equilibrium statistical mechanics.

The analysis of differential equations of such systems is often done approximately, using the WKB method ( also known as the Liouville – Green method ).

This is a special case of the general problem of Sturm – Liouville theory.

where is a Sturm – Liouville operator ( However it should be noted this operator may in fact be of the form where w ( x ) is the weighting function with respect to which the eigenfunctions of are orthogonal ) in the x coordinate.

The general problem of this type is solved in Sturm – Liouville theory.

The Liouville function's Dirichlet inverse is the absolute value of the Mobius function.

The Lambert series for the Liouville function is

Her submission included the celebrated discovery of what is now known as the " Kovalevsky top ", which was subsequently shown ( by Liouville ) to be the only other case of rigid body motion, beside the tops of Euler and Lagrange, that is " completely integrable ".

Imposing this regularity in the solution Θ of the second equation at the boundary points of the domain is a Sturm – Liouville problem that forces the parameter λ to be of the form λ = ℓ ( ℓ + 1 ) for some non-negative integer with ℓ ≥ | m |; this is also explained below in terms of the orbital angular momentum.

In mathematics, the Liouville – Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.

The result used for Liouville numbers in the proof is effective in the way it applies the mean value theorem: but improvements ( to what is now the Thue-Siegel-Roth theorem ) were not.

The fact that the domain of a non-constant elliptic function f can not be C is what Liouville actually proved, in 1847, using the theory of elliptic functions.

This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville ( see below ), which is what is most frequently referred to in this context.

In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense.

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 preserving for a Hamiltonian system.

Not all closed-form expressions have closed-form antiderivatives ; this study forms the subject of differential Galois theory, which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed form antiderivatives.

Joseph Liouville (, ; 24 March 1809 – 8 September 1882 ) was a French mathematician.

The value of λ is not specified in the equation ; finding the values of λ for which there exists a non-trivial solution of () satisfying the boundary conditions is part of the problem called the Sturm – Liouville ( S – L ) problem.

This is a Liouville dynamical system if ξ and η are taken as φ < sub > 1 </ sub > and φ < sub > 2 </ sub >, respectively ; thus, the function Y equals

Liouville proved by analytical means that if there is an elementary solution g to the equation g ′

Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields

Therefore the existence of a complete solution of the Hamilton – Jacobi equation is by no means a characterization of complete integrability in the Liouville sense.