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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, Liouville''s theorem, proved by Joseph Liouville in 1850, is a rigidity theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of Rn, 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 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.