Harmonic shears of regular polygons by hypergeometric functions
type: Item
instance of: scholarly article
title: Harmonic shears of regular polygons by hypergeometric functions
zbMATH Open document ID: 0937.30006
author: Peter Duren
author: Kathy Driver
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-03-30T00:00:00Z
review text: A harmonic mapping \(f\) that maps the unit disk \(\mathbb D\) onto a domain \(\Omega\) in the complex plane has a canonical representation \(f=h+\bar{g}\), where \(h\) and \(g\) are analytic in \(\mathbb D\) and \(g(0)=0\). Its dilatation is \(\omega=g'/h'\). The function \(f\) is called the harmonic shear of the analytic function \(\phi=h-g\). The authors construct harmonic shears (with prescribed dilatations) of functions \(\phi\) that map \(\mathbb D\) onto regular polygons. They also construct minimal surfaces associated to harmonic shears and study the inverse question: what conformal maps have harmonic shears that map \(\mathbb D\) onto a regular polygon. The constructions involve hypergeometric functions.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C20
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 1389236
zbMATH Keywords: harmonic mapping
zbMATH Keywords: harmonic shear
zbMATH Keywords: hypergeometric function
zbMATH Keywords: minimal surface
zbMATH Keywords: dilatation
zbMATH Keywords: Schwarz-Cristoffel mapping
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6544
OpenAlex ID: W2076649074
cites work: Q5820721
cites work: Harmonic univalent functions
cites work: Q4828548
cites work: Geometric properties of harmonic shears
cites work: Q3813974
cites work: Q5815016
cites work: Q5611582
cites work: Q3267566
cites work: On the Fourier series of a step function
DOI: 10.1006/JMAA.1999.6544
Recommended article: Harmonic shears of slit and polygonal mappings
Recommended article: Geometric properties of harmonic shears
Recommended article: Harmonic shears of elliptic integrals
Recommended article: Q3502279
Recommended article: Harmonic shears and numerical conformal mappings
Recommended article: Regular solids and harmonic polynomials
Recommended article: On exact quadrature formulas for harmonic functions on polyhedra
Recommended article: Polytopes, invariants and harmonic functions
Recommended article: Properties of harmonic functions defined by shear construction
Recommended article: Q4252575
description: scientific article; zbMATH DE number 1389236