Schwarz-Christoffel Formula for Multiply Connected Domains

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450 V. Mityushev CMFT domains [1, 2, 3, 4, 5]. However, explicit formulae for the Schottky-Klein prime function are known only under the restriction (1). In the present paper, we follow the method presented in [15] which is based on the construction of the conformal mapping via exact solution of a Riemann-Hilbert boundary value problem [11, 13, 14, 16] and on the uniformly convergent Poincaré series [12] for the classical Schottky groups. As a result we explicitly obtain the Schwarz-Christoffel formula for arbitrary domains D without any geometric restriction. In order to construct the conformal mapping we solve the following Riemann-Hilbert problem [8] (2) Re[(t − a k )ψ(t)] = −1, |t − a k | = r k , k = 1, 2, . . . , n, for the function ψ(z) analytic in D and continuous in D ∪ ∂D except at the point singularities on the boundary ∂D prescribed below. 2. Preliminaries The clockwise orientation is taken on each L k , hence the boundary ∂D = ⋃ n k=1 L k has the domain D to the left. First, we normalize the required conformal mapping f : D → P by the condition f(∞) = ∞. Let the boundary of the domain P consist of n mutually disjoint polygons Γ k = f(L k ) with P lying to the left. Let the M k vertices of Γ k be denoted by w lk , l = 1, 2, . . . , M k , numbered clockwise around Γ k for each k = 1, 2, . . . , n. The corresponding vertex angles of the Γ k at the vertices w lk , measured from the exterior of P, are introduced as π(1+β lk ), where πβ lk is the turning of the tangent at w lk . The constants β lk satisfy the inequality −1 < β lk ≤ 1 and the relations (3) ∑M k l=1 β lk = 2, k = 1, 2, . . . , n. The prevertices are denoted by z lk with f(z lk ) = w lk . Our study is based on the fact established in [8] that the preSchwarzian (4) S(z) = f ′′ (z) f ′ (z) is one of the solutions of the Riemann-Hilbert problem (2) in the class of functions having prescribed singularities at the points z lk ∈ L k , l = 1, 2, . . . , M k , k = 1, 2, . . . , n, where (5) S(z) ∼ β lk z − z lk as z → z lk . In order to recover f(z) from S(z) one can integrate twice the relation (4) since S(z) = (ln f ′ (z)) ′ . The primitive function (6) ω(z) = ∫ z S(ζ) dζ
12 (2012), No. 2 Schwarz-Christoffel Formula for Multiply Connected Domains 451 yields the Schwarz-Christoffel integral (7) f(z) = ∫ z exp(ω(ζ)) dζ. The preSchwarzian and exp(ω(ζ)) were constructed by DeLillo et al. [8] by infinite sequences of iterated reflections that generate a series which absolutely converges under the restriction (1) (for details see [8] and the next section) ⎡ ⎤ (8) exp(ω(z)) = n∏ ∏M m ⎣ ∏ m=1 l=1 γ o∈O ′ m z − γ o (z lm ) z − γ o (a m ) ∏ γ e∈E ′ m z − γ e (z lm ) z − γ e (a m ) ⎦β lm . In the present paper, we follow another method based on the Riemann-Hilbert problem (2) in the class of functions satisfying the asymptotic formulae (5). General solution ψ(z) of the problem (2) contains n arbitrary real constants [15], say ξ 1 , . . . , ξ n , i.e. ψ(z) = ψ(z; ξ 1 , . . . , ξ n ). In order to construct the required functions S(z) and ω(z), we first construct the functions ψ(z; ξ 1 , . . . , ξ n ) and Ω(z; ξ 1 , . . . , ξ n ). Further, the arbitrary constants ξ 1 , . . . , ξ n are chosen in such a way that ψ(z; ξ 1 , . . . , ξ n ) yields S(z) and Ω(z; ξ 1 , . . . , ξ n ) yields ω(z). 3. Schottky group The inversion of z through the circle L k is given by z ∗ (k) = r k 2 z − a k + a k . It is known that if a function Φ(z) is analytic in the disk |z − a k | < r k and continuous in its closure, then Φ(z ∗ (k) ) is analytic in |z − a k| > r k and continuous in |z − a k | ≥ r k . Introduce the composition of successive inversions through the circles L k1 , L k2 , . . . , L kp ( ) ∗ (9) z(k ∗ pk p−1 ...k 1 ) := z(k ∗ p−1 ...k 1 ) . (k p) In the sequence k 1 , k 2 , . . . , k p no two neighboring numbers are equal. The number p is called the level of the mapping. When p is even, these are Möbius transformations. If p is odd, we have anti-Möbius transformations, i.e., Möbius transformations in z. Thus, these mappings can be written in the form (10a) (10b) γ j (z) = e jz + b j c j z + d j for p ∈ 2Z, γ j (z) = e jz + b j c j z + d j for p ∈ 2Z + 1,
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Schwarz-Christoffel Formula for Multiply Connected Domains

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