Liouville's theorem

Liouville's theorem is a special case of the following statement:

He is remembered particularly for Liouville's theorem, a nowadays rather basic result in complex analysis.

Liouville's theorem states that any bounded entire function must be constant.

In light of the power series expansion, it is not surprising that Liouville's theorem holds.

His most famous result is the extension of Liouville's theorem to bounded entire functions.

This shows that f is bounded and entire, so it must be constant, by Liouville's theorem.

Liouville's theorem states this measure is invariant under the Hamiltonian flow.

This violates Liouville's theorem and presents a physical paradox.

By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space.

It is enough to prove that h can be extended to an entire function, in which case the result follows by Liouville's theorem.

This is a result of Liouville's theorem.

For if it were it would be bounded everywhere, and therefore constant by Liouville's theorem.

By Liouville's theorem, any angle-preserving local (conformal) transformation is of this form.

In this case, uniqueness follows by Liouville's theorem.

Thus h can be extended to an entire bounded function which by Liouville's theorem implies it is constant.

The flows are symplectomorphisms and hence obey Liouville's theorem.

In fact, it was Cauchy who proved Liouville's theorem.

For instance, all Hamiltonian systems are volume-preserving because of Liouville's theorem.

A corollary of this is the Liouville's theorem:

By Liouville's theorem, two meromorphic functions with the same zeros and poles must be proportional to one another.

Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle.

The only bounded harmonic functions defined on the whole plane are constant functions by Liouville's theorem.

(this can be viewed as an algebraic analogue of Liouville's theorem in complex analysis)

The generalized phase-space velocity is divergenceless, enabling Liouville's theorem.