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 0 < | x − p q | < 1 q n . {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{n}}}.}

Liouville's constant, sometimes also called Liouville's number, is the real number defined by L=sum_(n=1)^infty10^(-n!)=0.110001000000000000000001... (OEIS A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a factorial n!, and zeros everywhere else.

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 for which there exists a …

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 …

2024年5月29日 · Liouville's constant was proved to be transcendental by Joseph Liouville in 1844 as a demonstration that there exist real numbers which are provably transcendental. This was the simplest of several such numbers that he constructed.

2024年1月24日 · Let x be an irrational number that is algebraic of degree n. Then there exists a constant c> 0 (which can depend on x) such that: for every pair p, q ∈ Z with q ≠ 0. Liouville numbers are transcendental. Let r1, r2, …, rk be the rational roots of a polynomial P of degree n that has x as a root. Since x is irrational, it does not equal any ri.

Liouville's Theorem states that: Every holomorphic* function for which there exists a positive number M such that |f(z)| ≤ M | f (z) | ≤ M for all z ∈C z ∈ C is constant. I'm using this to prove the Fundamental Theorem of Algebra and I understand that for the proof to be complete:

2020年3月26日 · Therefore, $f(z)/z$ is an entire function bounded by $\max \{C,M\}$ on $\Bbb{C}$ and by Liouville's theorem, $f(z) / z$ is a constant, $K$. So we have $f(z) = Kz$ and we are given $|Kz| \leq C |z|^\alpha$. Then $|K| \leq C|z|^{\alpha - 1}$ for all $z \neq 0$. Taking the limit as $|z| \rightarrow \infty$, we find $K = 0$.

Liouville constant, which is also defined formally, is the first transcendental (not algebraic) number. It is defined in Section 6 quite generally as the sum X∞ k=1 a k bk! for a finite sequence {a k} k∈N and b∈N. Based on this definition, we also intro-duced the so-called Liouville number as L= X∞ k=1 10−k! = 0. ...

5 天之前 · A Liouville number is a transcendental number which has very close rational number approximations. An irrational number is called a Liouville number if, for each , there exist integers and such that.