Garnett J.B., Marshall D.E. Harmonic Measure

Cambridge University Press, 2008. — 592 p. — (New Mathematical Monographs: 2).This book provides an enlightening survey of remarkable new results that have only recently been discovered in the past two decades about harmonic measure in the complex plane. 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.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.Jordan domains
Finitely connected domains
Potential theory
Extremal distance
Applications and reverse inequalities
Simply connected domains, part one
Bloch functions and quasicircles
Simply connected domains, part two
Infinitely connected domains
Rectifiability and quadratic expressions
Appendices.