**Student Mathematical Library**

Volume: 9;
2000;
118 pp;
Softcover

MSC: Primary 26; 30; 51; 52; 53;

Print ISBN: 978-0-8218-2636-2

Product Code: STML/9

List Price: $24.00

Individual Price: $19.20

**Electronic ISBN: 978-1-4704-1816-8
Product Code: STML/9.E**

List Price: $24.00

Individual Price: $19.20

# Inversion Theory and Conformal Mapping

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*David E. Blair*

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.

#### Table of Contents

# Table of Contents

## Inversion Theory and Conformal Mapping

- Cover Cover11 free
- Other titles in this series i2 free
- Title page iii4 free
- Contents vii8 free
- Preface ix10 free
- Classical inversion theory in the plane 112 free
- Linear fractional transformations 2738
- Advanced calculus and conformal maps 6374
- Conformal maps in the plane 7586
- Conformal maps in Euclidean space 8394
- The classical proof of Liouville’s theorem 95106
- When does inversion preserve convexity? 107118
- Bibliography 115126
- Index 117128 free
- Back Cover Back Cover1130

#### 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