Schwarz-Christoffel Formula for Multiply Connected Domains

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452 V. Mityushev CMFT where e j d j − b j c j = 1, j = 0, 1, 2, . . .. Here γ 0 (z) := z (identical mapping with the level p = 0), γ 1 (z) := z ∗ (1), . . . , γ n (z) := z ∗ (n) (n simple inversions, p = 1), γ n+1 (z) := z ∗ (12), γ n+2 (z) := z ∗ (13), . . . , γ n 2(z) := z ∗ (n,n−1) (n 2 − n double inversions, p = 2), γ n 2 +1(z) := z ∗ (121), . . . and so on. The set of the subscripts j of γ j is ordered in such a way that the level p is increasing. The functions (10) generate a Schottky group K. Thus, each element of K is presented in the form of the composition of inversions (9) or in the form of linearly ordered functions (10). Let K m be a subset of K such that the last inversion of each element of K m is different from z ∗ (m) , i.e. K m = {z ∗ (k pk p−1 ...k 1 ) : k p ≠ m}. The set K ′ m = {z ∗ (k pk p−1 ...k 1 ) : k 1 ≠ m} is introduced similarly. All elements γ j of the even levels generate a subgroup E of the group K. The set of the elements γ j of odd level K\E is denoted by O. Introduce the notation E m = E ∩K m , O m = O ∩ K m and E ′ m = E ∩ K ′ m, O ′ m = O ∩ K ′ m. Let us fix an inversion z(m) ∗ . Consider the transformation γ j (z) = ( γt −1 (z) ) ∗ (m) from E, where γt −1 have (11) is the inverse transformation to γ t ∈ O ′ m. Then from [15] we ζ − γ j (z) ζ − γ j (w) = z − γ t(ζ(m) ∗ ) w − γ t (ζ(m) ∗ ) · w − γ t(a m ) z − γ t (a m ) . Consider now a transformation γ j ∈ O and from E ′ m. Then [15] (12) ( ) γ s (z) = γ −1 j z(m) ∗ ( ) ζ − γj (w) = w − γ s(ζ(m) ∗ ) ζ − γ j (z) z − γ s (ζ(m) ∗ ) · z − γ s(a m ) w − γ s (a m ) .
12 (2012), No. 2 Schwarz-Christoffel Formula for Multiply Connected Domains 453 4. Reduction of the Riemann-Hilbert problem to functional equations It follows from [15] that the inhomogeneous Riemann-Hilbert problem (2) always has solutions. The general solution amounts of a particular solution of the inhomogeneous problem and a linear combination of n solutions of the homogeneous problem. In order to solve the problem (2) rewrite it in the form of the R-linear problem (13) (t − a k )ψ(t) = (t − a k )ψ k (t) − (t − a k )ψ k (t) − 1 + iξ k , |t − a k | = r k , k = 1, . . . , n. Here, ξ k are undetermined real constants, ψ k (z) is analytic in |z − a k | < r k , continuous in |z − a k | ≤ r k except the points z lk , where (14) ψ k (z) ∼ β lk 2(z − z lk ) as z → z lk . It will be shown below in Lemma 1 that the asymptotics (14) and (5) for ψ(z) are matched. Hence, the R-linear problem (13) must be stated in a class of functions with prescribed singularities. It is convenient to describe this class by introduction of the Banach spaces as follows. Let G be a domain on the extended complex plane. Introduce the Banach space C(∂G) of functions continuous on ∂G with the norm ‖F ‖ = max |F (t)|. t∈∂G Let us consider a closed subspace C A (G) of C(∂G) consisting of functions analytically continued into G. The Maximum Principle implies that convergence in the space C A (G) is equivalent to uniform convergence in the closure of G. Let a fixed function F 0 have a finite number of singularities on the boundary ∂G. Introduce the space of functions endowed with the norm C A (G, F 0 ) = {F : F − F 0 ∈ C A (G)} ‖F ‖ CA (G,F 0 ) = max t∈∂G |F (t) − F 0(t)|. The spaces C A (G, F 0 ) and C A (G) are isomorphic. Introduce mutually disjointed disks D k = {z ∈ C: |z − a k | < r k }, k = 1, 2, . . . , n. The multiply connected domain D complements all the closed disks D k ∪ L k to the extended complex plane Ĉ = C ∪ {∞}, i.e. n D = Ĉ\ ⋃ (D k ∪ L k ). k=1
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