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• catalog abstract ""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 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 advanced undergraduate level and is suitable for a capstone course, topics course, senor 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."--Jacket.".
• catalog contributor b11815299.
• catalog created "2000.".
• catalog date "2000".
• catalog date "2000.".
• catalog dateCopyrighted "2000.".
• catalog description ""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 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 advanced undergraduate level and is suitable for a capstone course, topics course, senor 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."--Jacket.".
• catalog description "Chapter 1. Classical Inversion Theory in the Piane 1 -- 1.2. Cross ratio 9 -- 1.3. Applications 14 -- 1.4. Miquel's Theorem 17 -- 1.5. Feuerbach's Theorem 21 -- Chapter 2. Linear Fractional Transformations 27 -- 2.1. Complex numbers 27 -- 2.2. Extended complex plane and stereographic projection 29 -- 2.3. Linear fractional transformations 34 -- 2.4. Cross ratio 37 -- 2.5. Some special linear fractional transformations 39 -- 2.6. Extended Mobius transformations 43 -- 2.7. Poincare models of hyperbolic geometry 52 -- 2.8. A distortion theorem 59 -- Chapter 3. Advanced Calculus and Conformal Maps 63 -- 3.1. Review of advanced calculus 63 -- 3.2. Inner products 70 -- 3.3. Conformal maps 73 -- Chapter 4. Conformal Maps in the Plane 75 -- 4.1. Complex function theory 75 -- 4.2. Abundance of conformal maps 78 -- Chapter 5. Conformal Maps in Euclidean Space 83 -- 5.1. Inversion in spheres 83 -- 5.2. Conformal maps in Euclidean space 87 -- 5.3. Sphere preserving transformations 92 -- Chapter 6. Classical Proof of Liouville's Theorem 95 -- 6.1. Surface theory 95 -- 6.2. Classical proof 103 -- Chapter 7. When Does Inversion Preserve Convexity? 107 -- 7.1. Curve theory and convexity 107 -- 7.2. Inversion and convexity 110 -- 7.3. Problem for convex bodies 114.".
• catalog description "Includes bibliographical references and index.".
• catalog extent "x, 118 p. :".
• catalog identifier "0821826360 (softcover : alk. paper)".
• catalog isPartOf "Student mathematical library, 1520-9121 ; v. 9".
• catalog issued "2000".
• catalog issued "2000.".
• catalog language "eng".
• catalog publisher "Providence, RI : American Mathematical Society,".
• catalog subject "516.3/6 21".
• catalog subject "Conformal mapping.".
• catalog subject "Inversions (Geometry)".
• catalog subject "QA473 .B57 2000".
• catalog tableOfContents "Chapter 1. Classical Inversion Theory in the Piane 1 -- 1.2. Cross ratio 9 -- 1.3. Applications 14 -- 1.4. Miquel's Theorem 17 -- 1.5. Feuerbach's Theorem 21 -- Chapter 2. Linear Fractional Transformations 27 -- 2.1. Complex numbers 27 -- 2.2. Extended complex plane and stereographic projection 29 -- 2.3. Linear fractional transformations 34 -- 2.4. Cross ratio 37 -- 2.5. Some special linear fractional transformations 39 -- 2.6. Extended Mobius transformations 43 -- 2.7. Poincare models of hyperbolic geometry 52 -- 2.8. A distortion theorem 59 -- Chapter 3. Advanced Calculus and Conformal Maps 63 -- 3.1. Review of advanced calculus 63 -- 3.2. Inner products 70 -- 3.3. Conformal maps 73 -- Chapter 4. Conformal Maps in the Plane 75 -- 4.1. Complex function theory 75 -- 4.2. Abundance of conformal maps 78 -- Chapter 5. Conformal Maps in Euclidean Space 83 -- 5.1. Inversion in spheres 83 -- 5.2. Conformal maps in Euclidean space 87 -- 5.3. Sphere preserving transformations 92 -- Chapter 6. Classical Proof of Liouville's Theorem 95 -- 6.1. Surface theory 95 -- 6.2. Classical proof 103 -- Chapter 7. When Does Inversion Preserve Convexity? 107 -- 7.1. Curve theory and convexity 107 -- 7.2. Inversion and convexity 110 -- 7.3. Problem for convex bodies 114.".
• catalog title "Inversion theory and conformal mapping / David E. Blair.".
• catalog type "text".