Softcover ISBN:  9780821826362 
Product Code:  STML/9 
List Price:  $49.00 
Individual Price:  $39.20 
eBook ISBN:  9781470418168 
Product Code:  STML/9.E 
List Price:  $39.00 
Individual Price:  $31.20 
Softcover ISBN:  9780821826362 
eBook: ISBN:  9781470418168 
Product Code:  STML/9.B 
List Price:  $88.00 $68.50 
Softcover ISBN:  9780821826362 
Product Code:  STML/9 
List Price:  $49.00 
Individual Price:  $39.20 
eBook ISBN:  9781470418168 
Product Code:  STML/9.E 
List Price:  $39.00 
Individual Price:  $31.20 
Softcover ISBN:  9780821826362 
eBook ISBN:  9781470418168 
Product Code:  STML/9.B 
List Price:  $88.00 $68.50 

Book DetailsStudent Mathematical LibraryVolume: 9; 2000; 118 ppMSC: Primary 26; 30; 51; 52; 53;
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 circlepreserving maps. A particular feature is the inclusion of a paper by Carathéodory with the remarkable result that any circlepreserving 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.ReadershipAdvanced undergraduate students and mathematicians interested in conformal mappings in higherdimensional spaces.

Table of Contents

Chapters

Chapter 1. Classical inversion theory in the plane

Chapter 2. Linear fractional transformations

Chapter 3. Advanced calculus and conformal maps

Chapter 4. Conformal maps in the plane

Chapter 5. Conformal maps in Euclidean space

Chapter 6. The classical proof of Liouville’s theorem

Chapter 7. When does inversion preserve convexity?


Additional Material

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 wellwritten 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


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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 circlepreserving maps. A particular feature is the inclusion of a paper by Carathéodory with the remarkable result that any circlepreserving 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.
Advanced undergraduate students and mathematicians interested in conformal mappings in higherdimensional spaces.

Chapters

Chapter 1. Classical inversion theory in the plane

Chapter 2. Linear fractional transformations

Chapter 3. Advanced calculus and conformal maps

Chapter 4. Conformal maps in the plane

Chapter 5. Conformal maps in Euclidean space

Chapter 6. The classical proof of Liouville’s theorem

Chapter 7. When does inversion preserve convexity?

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 wellwritten 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