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Inversion Theory and Conformal Mapping
David E. Blair, Michigan State University, East Lansing, MI
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Student Mathematical Library
2000; 118 pp; softcover
Volume: 9
ISBN-10: 0-8218-2636-0
ISBN-13: 978-0-8218-2636-2
List Price: US$22
Member Price: US$17.60
Order Code: STML/9
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It 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 Carathéodory with the remarkable result that any circle-preserving transformation is necessarily a Möbius 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.

M. C. Escher's Hand with Reflecting Sphere ©2000 Cordon Art B.V. - Baarn - Holland. All rights reserved.

Readership

Advanced undergraduate students and mathematicians interested in conformal mappings in higher-dimensional spaces.

Reviews

"Gives several beautiful applications ... reprints a wonderful paper of C. Carathéodory ... a very nicely written book with interesting results on almost every page. It should be very useful as the basis of an advanced undergraduate capstone course, or as a supplement to more standard material, or simply to sit and read for the entertainment and enlightenment it offers."

-- Mathematical Reviews

"A very well-written and intriguing book ... Anyone who is interested in inversion theory and conformal mapping should have this book in his personal library. [It] can be used as an excellent reference book for a graduate course. It can also be used as a textbook for an advanced undergraduate course, capstone course, topics course, senior seminar or independent study."

-- MAA Online

Table of Contents

  • Classical 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
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