Harmonic Mappings in the Plane

Peter Duren - Collection Cambridge Tracts in Mathematics

212 pages, parution le 04/06/2004

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Résumé

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.

Sommaire

Preliminaries
General properties of Harmonic Mappings
Harmonic Mappings onto Convex Regions
Harmonic Self-Mappings of the Disk
Harmonic Univalent Functions
Extremal Problems
Mapping Problems
Additional Topics
Minimal Surfaces
Curvature of Minimal Surfaces

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	PAPIER
Éditeur(s)	Cambridge University Press
Auteur(s)	Peter Duren
Collection	Cambridge Tracts in Mathematics
Parution	04/06/2004
Nb. de pages	212
Format	15,5 x 23,5
Couverture	Relié
Poids	450g
Intérieur	Noir et Blanc
EAN13	9780521641210
ISBN13	978-0-521-64121-0