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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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Lenguaje: | eng |
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American Mathematical Society
2000
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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. |
id | cern-2623064 |
institution | Organización Europea para la Investigación Nuclear |
language | eng |
publishDate | 2000 |
publisher | American Mathematical Society |
record_format | invenio |
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 |