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.

Harmonic Mappings onto Convex Domains

1987 ◽  
Vol 39 (6) ◽  
pp. 1489-1530 ◽  
Author(s):  
Yusuf Abu-Muhanna ◽  
Glenn Schober
Keyword(s):  
Univalent Functions ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Convex Domains ◽  
Harmonic Solution ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected ◽  
Complex Valued

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equationwhere a is analytic and |a| < 1, such that f(U) ⊂ D and

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

scholarly journals Asymptotic tracts of harmonic functions II

1995 ◽  
Vol 38 (1) ◽  
pp. 35-52 ◽  
Author(s):  
K. F. Barth ◽  
D. A. Brannan
Keyword(s):  
Harmonic Function ◽  
Real Number ◽  
Real Function ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected ◽  
Prime End

An asymptotic tract of a real function u harmonic and non-constant in ℂ is a component of the set {z:u(z)≠c}, for some real number c; a quasi-tractT(≠ℂ) is an unbounded simply-connected domain in ℂ such that there exists a function u that is positive, unbounded and harmonic in T such that, for each point ζ∈∂T∩ℂ,and a ℱ-tract is an unbounded simply-connected domain T in ℂ whose every prime end that contains ∞ in its impression is of the first kind.The authors study the growth of a harmonic function in one of its asymptotic tracts, and the question of whether a quasi-tract is an asymptotic tract. The branching of either type of tract is also taken into consideration.

scholarly journals Asymptotic Solution Of Differential Equations In a Domain Containing a Regular Singular Point

1956 ◽  
Vol 8 ◽  
pp. 97-104 ◽  
Author(s):  
N. D. Kazarinoff ◽  
R. McKelvey
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Singular Point ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Connected Domain ◽  
Regular Singular Point ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

1. Introduction. In this paper we study the asymptotic behavior in λ of the solutions about the origin in the z-plane of the differential equation.Both the variable z and the parameter λ are complex. The coefficient P(z, λ) is assumed to be analytic and single-valued in λ at infinity and in z throughout a bounded, closed, simply connected domain D containing z = 0.

scholarly journals A characterization of the algebra of holomorphic functions on a simply connected domain

1989 ◽  
Vol 12 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Derming Wang ◽  
Saleem Watson
Keyword(s):  
Connected Domain ◽  
Holomorphic Functions ◽  
Simply Connected Domain ◽  
Simply Connected

Let A be a singly-generated ℱ-algebra. It is shown that A isomorphic to H(Ω) where Ω is a simply connected domain in ℂ if and only if A has no topological divisors of zero. It follows from this that there are exactly three ℱ-algebras (up to isomorphism) which are singly generated and have no topological divisors of zero.

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.

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