Abstract
Given a conformal mapping $f$ of the unit disk $\mathbb D$ onto a simply connected domain $D$ in the complex plane bounded by a closed Jordan curve, we consider the problem of constructing a matching conformal mapping, i.e., the mapping of the exterior of the unit disk $\mathbb D^*$ onto the exterior domain $D^*$ regarding to $D$. The answer is expressed in terms of a linear differential equation with a driving term given as the kernel of an operator dependent on the original mapping $f$. Exampl
Results & Lemmas (6)
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Theorem 1.
Theorem 1. Suppose f ∈S1,α and ϕ, ϕ(∞) = ∞, are matching univalent functions. Then the kernel of the operator If: Lipα(S1, R) →Hol(D) is…
Theorem 1. Suppose f ∈S1,α and ϕ, ϕ(∞) = ∞, are matching univalent functions. Then the kernel of the operator If : Lipα(S1, R) →Hol(D) is the 3
Theorem 1
Theorem 1 describes the real-valued solutions to the equation If[v] = 0. The set of complex solutions to this equation is much more…
Theorem 1 describes the real-valued solutions to the equation If[v] = 0. The set of complex solutions to this equation is much more extensive. Denote by HolC(D∗) the class of all continuous functions h : D∗∪S1 →C which are analytic in D∗.
Theorem 2.
Theorem 2. Suppose f ∈S1,α and ϕ, ϕ(∞) = ∞, are matching univalent functions, and γ:= f −1 ◦ϕ is the induced homeomorphism of S1. Then the…
Theorem 2. Suppose f ∈S1,α and ϕ, ϕ(∞) = ∞, are matching univalent functions, and γ := f −1 ◦ϕ is the induced homeomorphism of S1. Then the kernel of the operator If : Lipα(S1, C) →Hol(D) coincides with the set of all functions v of the form v(z) = v0(z) · (h ◦γ−1)(z), z ∈S1, (6) where h is an arbitrary function belonging to HolC(D∗) ∩Lipα(S1, C) and v0 is defined by (4). In Section 2 we show how the operator If appears in a natural way within the identification of the Kirillov’s homogeneous manif
Proposition 1.
Proposition 1. The kernel of If: F →Hol(D) is one-dimensional and coincides with span 1/(γ−1)#. 7
Proposition 1. The kernel of If : F →Hol(D) is one-dimensional and coincides with span{1/(γ−1)#}. 7
Proposition 2.
Proposition 2. The complex structure on TM induced by the standard com- plex structure on FC via If is given by Jγ = AdγJ0 (Adγ)−1, where…
Proposition 2. The complex structure on TM induced by the standard com- plex structure on FC via If is given by Jγ = AdγJ0 (Adγ)−1, where Adγ stands for the differential of Aγβ := γ ◦β ◦γ−1 at the origin β = id.
Proposition 3.
Proposition 3. Suppose γ ∈Diff+(S1) is such that v0:= 1/(γ−1)# is of the form (16). Then the function f ∈S∞that corresponds to γ via con-…
Proposition 3. Suppose γ ∈Diff+(S1) is such that v0 := 1/(γ−1)# is of the form (16). Then the function f ∈S∞that corresponds to γ via con- formal welding, is a solution to differential equation (18) with b0 := 1 and wk := f(zk). Moreover, the vector (w1, . . . , wn) satisfies system (19), (20), provided all the roots zk of Q are simple.