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Geometric Properties of Harmonic Shears

Geometric Properties of Harmonic Shears This paper is a study of planar harmonic mappings produced with the “shear construction” devised by Clunie and Sheil-Small in 1984. Specifically it will describe the geometry of mappings produced by the shear construction. The first section introduces basic concepts, including a description of the shear construction itself, a technique for constructing examples of harmonic mappings by shearing a conformal mapping. The main body of the paper alternates between presenting examples of harmonic shears, illustrating them graphically with the help of Mathematica software, and using an integral representation of the harmonic shear to explain and predict characteristics of the image domain. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

Geometric Properties of Harmonic Shears

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Publisher
Springer Journals
Copyright
Copyright © 2004 by Heldermann  Verlag
Subject
Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/BF03321057
Publisher site
See Article on Publisher Site

Abstract

This paper is a study of planar harmonic mappings produced with the “shear construction” devised by Clunie and Sheil-Small in 1984. Specifically it will describe the geometry of mappings produced by the shear construction. The first section introduces basic concepts, including a description of the shear construction itself, a technique for constructing examples of harmonic mappings by shearing a conformal mapping. The main body of the paper alternates between presenting examples of harmonic shears, illustrating them graphically with the help of Mathematica software, and using an integral representation of the harmonic shear to explain and predict characteristics of the image domain.

Journal

Computational Methods and Function Theory – Springer Journals

Published: Mar 7, 2013

References

  • Boundary values versus dilatations of harmonic mappings
    Bshouty, D; Hengartner, W
  • Harmonic univalent functions
    Clunie, J; Sheil-Small, T
  • Boundary correspondence and dilatation of harmonic mappings
    Duren, P; Khavinson, D
  • On the boundary behavior of orientation-preserving harmonic mappings
    Hengartner, W; Schober, G
  • Harmonic mappings with given dilatation
    Hengartner, W; Schober, G
  • Planar harmonic maps with inner and Blaschke dilatations
    Laugesen, R S