# Liouville theorems

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Liouville's theorem on bounded entire analytic functions: If an entire function of the complex variable is bounded, that is, then is a constant. This proposition, which is one of the fundamental results in the theory of analytic functions, was apparently first published in 1844 by A.L. Cauchy

for the case ; J. Liouville presented it in his lectures in 1847, and this is how the name arose.

Liouville's theorem can be generalized in various directions. For example, if is an entire function in and for some integer , then is a polynomial in the variables of degree not exceeding . Moreover, if is a real-valued harmonic function in the number space , , and  , then is a harmonic polynomial in of degree not exceeding (see also ).

Liouville's theorem on conformal mapping: Every conformal mapping of a domain in a Euclidean space with can be represented as a finite number of compositions of very simple mappings of four kinds — translation, similarity, orthogonal transformation, and inversion. It was proved by J. Liouville in 1850 (see , Appendix 6).

This Liouville theorem shows the poverty of the class of conformal mappings in space, and from this point of view it is very important in the theory of analytic functions of several complex variables and in the theory of quasi-conformal mapping.

How to Cite This Entry:
Liouville theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Liouville_theorems&oldid=19033
This article was adapted from an original article by E.D. Solomentsev, S.A. Stepanov, I.A. Kvasnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article