# Liouville's theorem (conformal mappings)

- Liouville's theorem (conformal mappings)
In mathematics, **Liouville's theorem** is a theorem about conformal mappings in Euclidean space. It states that any conformal mapping on a domain of **R**^{"n"}, where "n" > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are all Möbius transformations. This severely limits the variety of possible conformal mappings in **R**^{3} and higher-dimensional spaces.

By contrast, conformal mappings in **R**^{2} can be much more complicated - for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.

**Bibliography**

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*citation|first=Gaspard|last=Monge|authorlink=Gaspard Monge|title=Application de l'analyse à la Géométrie|year=1850|publisher=Bachelier|pages=609–616|url=http://books.google.com/books?id=iCEOAAAAQAAJ&dq=%22Application+de+l%27analyse+%C3%A0+la+g%C3%A9om%C3%A9trie%22

*springer|id=L/l059680|title=Liouville theorems|first=E.D.|last=Solomentsev|year=2001

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