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.

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 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.

scholarly journals On harmonic continuation

1963 ◽  
Vol 6 (1) ◽  
pp. 54-56
Author(s):  
M. S. P. Eastham
Keyword(s):  
Analytic Function ◽  
Finite Number ◽  
Connected Domain ◽  
Arc Length ◽  
Jordan Curves ◽  
Simply Connected Domain ◽  
Harmonic Continuation ◽  
Simply Connected

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.

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