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OverviewIt is rarely taught in undergraduate or even graduate curricula 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. This fact is taught in most complex analysis courses. 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, due to Nevanlinna, 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 Caratheodory with the remarkable result that any circle-preserving transformation is necessarily a Mobius 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 as an independent study text. 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. Full Product DetailsAuthor: David E. BlairPublisher: American Mathematical Society Imprint: American Mathematical Society Volume: No. 9 Dimensions: Width: 15.60cm , Height: 0.70cm , Length: 23.40cm Weight: 0.165kg ISBN: 9780821826362ISBN 10: 0821826360 Pages: 118 Publication Date: 30 August 2000 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Awaiting stock  The supplier is currently out of stock of this item. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out for you. Table of ContentsClassical inversion theory in the plane Linear fractional transformations Advanced calculus and conformal maps Conformal maps in the plane Conformal maps in Euclidean space The classical proof of Liouville's theorem When does inversion preserve convexity? Bibliography Index.ReviewsAuthor InformationTab Content 6Author Website:Countries AvailableAll regions |
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