Evaluation of a boundary integral representation for the conformal mapping of the unit disk onto a simply-connected domain

1995 ◽  
Vol 3 (1-2) ◽  
pp. 115-135 ◽  
Author(s):  
S. N. Chandler-Wilde ◽  
J. Levesley ◽  
D. M. Hough
Keyword(s):  
Conformal Mapping ◽  
Integral Representation ◽  
Unit Disk ◽  
Connected Domain ◽  
Boundary Integral ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Boundary Integral Representation

A Variation of the Koebe Mapping in a Dense Subset of S

1987 ◽  
Vol 39 (1) ◽  
pp. 54-73 ◽  
Author(s):  
D. Bshouty ◽  
W. Hengartner
Keyword(s):  
Conformal Mapping ◽  
Uniform Convergence ◽  
Connected Domain ◽  
Dual Space ◽  
Dense Subset ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let H(U) be the linear space of holomorphic functions defined on the unit disk U endowed with the topology of normal (locally uniform) convergence. For a subset E ⊂ H(U) we denote by Ē the closure of E with respect to the above topology. The topological dual space of H(U) is denoted by H′(U).Let D, 0 ∊ D, be a simply connected domain in C. The unique univalent conformal mapping ϕ from U onto D, normalized by ϕ(0) = 0 and ϕ′(0) > 0 will be called “the Riemann Mapping onto D”. Let S be the set of all normalized univalent functions

scholarly journals On some properties of solutions of the p-harmonic equation

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 577-591 ◽  
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Unit Disk ◽  
Connected Domain ◽  
Harmonic Mappings ◽  
Simply Connected Domain ◽  
Harmonic Equation ◽  
Simply Connected ◽  
Complex Valued ◽  
Class Of Functions

A 2p-times continuously differentiable complex-valued function ? = u + iv in a simply connected domain ? ? C is p-harmonic if ? satisfies the p-harmonic equation ?p? = 0. In this paper, we investigate the properties of p-harmonic mappings in the unit disk |z| < 1. First, we discuss the convexity, the starlikeness and the region of variability of some classes of p-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form D? = z?z - ??z, where f is p-harmonic in |z| < 1. Also, we discuss the region of variability for certain p-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for bi- and tri-harmonic mappings.

scholarly journals Relative Distance and Quasi-Conformal Mappings

1960 ◽  
Vol 16 ◽  
pp. 111-117
Author(s):  
D. A. Storvick
Keyword(s):  
Conformal Mapping ◽  
Unit Circle ◽  
Distance Function ◽  
Connected Domain ◽  
General Class ◽  
Conformal Mappings ◽  
Relative Distance ◽  
Simply Connected Domain ◽  
Simply Connected

1. Introduction. M. A. Lavrentiev made use of a relative distance function to establish some important results concerning the correspondence between the frontiers under a conformal mapping of a simply connected domain onto the unit circle. The purpose of this note is to show that some of these results are valid for the boundary correspondences induced by the more general class of quasi-conformal mappings.

scholarly journals A version of Runge's theorem for the Helmholtz equation with applications to scattering theory

1989 ◽  
Vol 32 (1) ◽  
pp. 107-119 ◽  
Author(s):  
R. L. Ochs
Keyword(s):  
Wave Function ◽  
Helmholtz Equation ◽  
Connected Domain ◽  
Scattering Theory ◽  
Polar Coordinates ◽  
Differentiable Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

Let D be a bounded, simply connected domain in the plane R2 that is starlike with respect to the origin and has C2, α boundary, ∂D, described by the equation in polar coordinateswhere C2, α denotes the space of twice Hölder continuously differentiable functions of index α. In this paper, it is shown that any solution of the Helmholtz equationin D can be approximated in the space by an entire Herglotz wave functionwith kernel g ∈ L2[0,2π] having support in an interval [0, η] with η chosen arbitrarily in 0 > η < 2π.

Alternative Boundary-Integral Representations of Ship Waves

2002 ◽  
Author(s):  
Francis Noblesse ◽  
Chi Yang ◽  
Dane Hendrix ◽  
Rainald Lo¨hner
Keyword(s):  
Integral Representation ◽  
Fundamental Problem ◽  
Boundary Integral ◽  
Ship Hull ◽  
Integral Representations ◽  
Singular Boundary ◽  
Ship Waves ◽  
Classical Boundary ◽  
The Green Function ◽  
Boundary Integral Representation

The fundamental problem of determining the free-surface potential flow that corresponds to a given flow at a ship hull surface is reconsidered. Stokes’ theorem is used to transform the dipole distribution over the ship hull surface in the classical boundary-integral representation of the velocity potential. This Stokes’ transformation yields a weakly-singular boundary-integral representation that defines the potential in terms of the Green function G and related functions that are no more singular than G. Accordingly, the velocity representation only involves functions that are no more singular than ∇G.

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