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Harmonic Mappings in the Plane
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Harmonic mappings in the plane are univalent complex-valued harmonic functions of a complex variable. Conformal mappings are a special case where the real and imaginary parts are conjugate harmonic functions, satisfying the Cauchy-Riemann equations. Harmonic mappings were studied classically by differential geometers because they provide isothermal (or conformal) parameters for minimal surfaces. More recently they have been actively investigated by complex analysts as generalizations of univalent analytic functions, or conformal mappings. Many classical results of geometric function theory extend to harmonic mappings, but basic questions remain unresolved. This book is the first comprehensive account of the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.  Essentially self-contained, the book contains background material in complex analysis and a full development of the classical theory of minimal surfaces, including the Weierstrass-Enneper representation. It is designed to introduce non-specialists to a beautiful area of complex analysis and geometry.

Reviews

‘This book is devoted to the theory of planar harmonic mappings, treating both the generalizations of univalent analytic functions and the connections with minimal surfaces.‘

Source: Monatshefte für Mathematik

‘For all students in this filed Duren's book will be essential reading. it will also be the classic reference book in this area.‘

Source: Proceedings of the Edinburgh Mathematical Society

'Those who are sensible to the beauty of complex functions and Riemann surfaces will certainly enjoy reading this nicely written … book.'

Source: Mathematical Geology

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Contents

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