# Liouville's Theorem

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## Liouville's theorem

[′lyü‚vēlz ‚thir·əm]
(mathematics)
Every function of a complex variable which is bounded and analytic in the entire complex plane must be constant.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Liouville’s Theorem

(1) In mechanics, a theorem asserting that the volume in phase space of a system obeying the equations of mechanics in Hamiltonian form remains constant as the system moves. Liouville’s theorem was established in 1838 by the French scientist J. Liouville.

The state of a mechanical system defined by the generalized coordinates q1, q2, . . ., qN and the canonically conjugate generalized momenta p1, p2, . . ., pN (where N is the number of degrees of freedom of the system) can be considered as a point with rectangular Cartesian coordinates q1, q2, ..., qN; p1, p2, . . ., pN is a 2N-dimensional space called the phase space. The evolution of the system in time is represented as the motion of this phase point in the 27V-dimensional space. If phase points entirely fill some region of the phase space at the initial moment of time and pass over into another region of the space in the course of time, then the corresponding phase volume, according to Liouville’s theorem, will be the same. Thus the motion of the points that represent the state of the system in the phase space resembles that of an incompressible fluid.

Liouville’s theorem permits introduction of a distribution function of the particles of the system in phase space and is the basis of statistical physics.

### REFERENCES

Synge, J. L. Klassicheskaia dinamika. Moscow, 1963. (Translated from English.)
Gibbs, J. Osnovnye printsipy statisticheskoi mekhaniki. Moscow, 1946. (Translated from English.)
Leontovich, M. A. Statisticheskaia fizika. Moscow-Leningrad, 1944.
D. N. ZUBAREV
(2) In the theory of analytic functions, a theorem asserting that every entire function that is finite on the entire complex plane is identically a constant. Liouville’s theorem is named after J. Liouville, who made it the basis of his lectures (1847) on the theory of elliptical functions. However, it was first formulated and proved by A. Cauchy in 1844.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
By Liouville's theorem, the boundedness of [[absolute value of u].sup.2] (z) [[absolute value of v].sup.2](z) implies that there exists a constant C such that uv = C.
Since the trace of P(t, s) is equal to - [f'.sup.x] (t, [x.sub.0](t)), applying Liouville's theorem, we have
Furthermore Mahler's version of Liouville's Theorem says that if [alpha] [member of] K(([X.sup.-1])) is algebraic over K(X) of degree n > 1 then B([alpha], n - 1) = 0.
The 13 papers consider such topics as the fundamental matrix solution for higher-order parabolic systems, variations on Liouville's theorem in the setting of stationary flows of generalized Newtonian fluids in the plane, the boundary Harnack principle for second-order elliptic equations in John domains and uniform domains, oblique derivative problems for non-divergence parabolic equations with time-discontinuous coefficients, and the local regularity theory for the Navier-Stokes equations near the boundary.
Hence, by Liouville's theorem, [psi](z) [equivalent to] 0, and then (8) implies that K = sinc.
For an nonconstant entire function f, we must have [[rho].sub.log](f) [greater than or equal to] 1, by the usual proof of Liouville's theorem. It is easily seen that if f(z) has logarithmic order p then so has the function f(az + b) for a [not equal to] 0.
By the maximum modulus principle [12] the entire function F(z) is bounded on the complex plane C and by Liouville's theorem [12] we conclude that F(z) is a constant function, i.e., there is a constant [kappa] such that
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