Liouville's theorem (conformal mappings)

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In mathematics, Liouville's theorem is a theorem about conformal mappings in Euclidean space. It states that any conformal mapping on a domain of Rⁿ, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions. This severely limits the variety of possible conformal mappings in R³ and higher-dimensional spaces.

By contrast, conformal mappings in R² can be much more complicated - for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem.

[edit] Bibliography

• Springer Encyclopedia of Mathematics, ISBN 1402006098, also available online at [1]
• David E. Blair (2000) Inversion Theory and Conformal Mapping, American Mathematical Society, ISBN 0821826360, Chapter 6, "The Classical Proof of Liouville's Theorem", pp.95 - 105.