Vol. 28, No. 2, 1969

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Liouville’s theorem

Robert Phillips

Vol. 28 (1969), No. 2, 397–405

Liouville’s theorem states that in Euclidean space of dimension greater than two, every conformal mapping must, by necessity, be an elementary transformation (i.e., a translation, a magnification, an orthogonal transformation, a reflection through reciprocal radii, or a combination of these transformations). This theorem was proven by R. Nevanlinna under the additional assumption that the mappings be at least four times differentiable. We show that a modified version of Nevanlinna’s proof is still valid when the mappings are assumed to be only twice differentiable. Our methods are those of Nonstandard Analysis as developed by A. Robinson.

Mathematical Subject Classification
Primary: 30.40
Received: 28 March 1968
Published: 1 February 1969
Robert Phillips