During the last two decades several remarkable new results were discovered about harmonic measure in the complex plane. This book provides a careful survey of these results and an introduction to the branch of analysis which contains them. Many of these results, due to Bishop, Carleson, Jones, Makarov, Wolff and others, appear here in paperback for the first time. The book is accessible to students who have completed standard graduate courses in real and complex analysis. The first four chapters provide the needed background material on univalent functions, potential theory, and extremal length, and each chapter has many exercises to further inform and teach the readers.Read more
- Early chapters designed for a second-year graduate course in complex analysis; later chapters bring the reader up to the cutting-edge of research
- Picks up where other books leave off in recent intensive investigations in complex analysis
- Nearly 200 exercises and over 100 diagrams aid understanding and allow the reader to test their knowledge
Reviews & endorsements
'The authors say that they wrote this book to explain the exciting new developments in the subject over the past couple of decades. They have achieved this with an impressive scholarship and outstanding expository clarity. … it has clearly been written with great care and deep insight into the subject matter. I expect it to become a highly valued addition to the bookshelves of students and experienced researchers alike.' Zentralblatt MATHSee more reviews
'… everybody who is interested in function theory and for whom Harmonic Measure sounds somewhat familiar and potentially interesting will find this book extremely useful, wonderfully well written and a joy to read.' MAA Reviews
'Over the last 20 years I have often been asked to suggest a 'good place to learn about harmonic measure,' and from now on the book of Garnett and Marshall will be my first suggestion. It's a great place for graduate students to learn an important area from the foundations up to the research frontier or for experts to locate a needed result or reference … The book is well organized and well written ... It deserves a large audience because this material is fundamental to modern complex analysis and has important connections to probability, dynamics, functional analysis and other areas. It will be of immense value to both expert practitioners and students. This is one of a handful of books I keep on my desk (rather than up on a shelf), and I often look through its pages to educate or entertain myself. It is an illuminating survey of the geometric theory of harmonic measure as it stands today and is sure to become a respected textbook and standard reference that will profoundly influence the future development of the field.' Bulletin of the American Mathematical Society
'… can be warmly recommended to students and researchers with a deep interest in analysis. It is an excellent preparation for serious work in complex analysis or potential theory.' EMS Newsletter
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- Date Published: June 2008
- format: Paperback
- isbn: 9780521720601
- length: 592 pages
- dimensions: 228 x 152 x 28 mm
- weight: 0.8kg
- contains: 126 b/w illus. 1 table 190 exercises
- availability: Available
Table of Contents
1. Jordan domains
2. Finitely connected domains
3. Potential theory
4. Extremal distance
5. Applications and reverse inequalities
6. Simply connected domains, part one
7. Bloch functions and quasicircles
8. Simply connected domains, part two
9. Infinitely connected domains
10. Rectifiability and quadratic expressions
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