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Inversion theory and conformal mapping

It is rarely taught in an undergraduate or even graduate curriculum that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. The principal aim o...

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Detalles Bibliográficos
Autor principal: Blair, David E
Lenguaje:eng
Publicado: American Mathematical Society 2000
Materias:
Acceso en línea:http://cds.cern.ch/record/2623064
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author Blair, David E
author_facet Blair, David E
author_sort Blair, David E
collection CERN
description It is rarely taught in an undergraduate or even graduate curriculum that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof in general dimension and a differential-geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Carath�odory with the remarkable result that any circle-preserving transformation is necessarily a M�bius transformation, not even the continuity of the transformation is assumed. The text is at the level of advanced undergraduates and is suitable for a capstone course, topics course, senior seminar or independent study. Students and readers with university courses in differential geometry or complex analysis bring with them background to build on, but such courses are not essential prerequisites.
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spelling cern-26230642021-04-21T18:47:46Zhttp://cds.cern.ch/record/2623064engBlair, David EInversion theory and conformal mappingMathematical Physics and MathematicsIt is rarely taught in an undergraduate or even graduate curriculum that the only conformal maps in Euclidean space of dimension greater than two are those generated by similarities and inversions in spheres. This is in stark contrast to the wealth of conformal maps in the plane. The principal aim of this text is to give a treatment of this paucity of conformal maps in higher dimensions. The exposition includes both an analytic proof in general dimension and a differential-geometric proof in dimension three. For completeness, enough complex analysis is developed to prove the abundance of conformal maps in the plane. In addition, the book develops inversion theory as a subject, along with the auxiliary theme of circle-preserving maps. A particular feature is the inclusion of a paper by Carath�odory with the remarkable result that any circle-preserving transformation is necessarily a M�bius transformation, not even the continuity of the transformation is assumed. The text is at the level of advanced undergraduates and is suitable for a capstone course, topics course, senior seminar or independent study. Students and readers with university courses in differential geometry or complex analysis bring with them background to build on, but such courses are not essential prerequisites.American Mathematical Societyoai:cds.cern.ch:26230642000
spellingShingle Mathematical Physics and Mathematics
Blair, David E
Inversion theory and conformal mapping
title Inversion theory and conformal mapping
title_full Inversion theory and conformal mapping
title_fullStr Inversion theory and conformal mapping
title_full_unstemmed Inversion theory and conformal mapping
title_short Inversion theory and conformal mapping
title_sort inversion theory and conformal mapping
topic Mathematical Physics and Mathematics
url http://cds.cern.ch/record/2623064
work_keys_str_mv AT blairdavide inversiontheoryandconformalmapping