Schwarz-Christoffel Formula for Multiply Connected Domains

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456 V. Mityushev CMFT i.e. χ k ∈ C A (D k ). Substitution of (21) into (19) yields the system of functional equations in C A (D k ), k = 1, 2, . . . , n, χ k (z) = − ∑ ( ) 2 rm ) (22) χ m (z(m) ∗ + h k (z), z − a m m≠k where (23) h k (z) = − ∑ m≠k |z − a k | ≤ r k , k = 1, . . . , n, 1 − iξ m + Φ(z) − Φ k (z) − ∑ ( ) 2 rm ) Φ m (z(m) ∗ z − a m z − a m m≠k belongs to C A (D k ) (see (15) and (16)). It follows from [16, 14] that the system of functional equations (22) has a unique solution in C A (D k ). This solution χ k (z) can be found by the method of uniformly convergent successive approximations. Then (21) yields ψ k (z) = Φ k (z) + χ k (z). Hence, ψ k (z) can be found by the method of successive approximations applied to (19). Convergence of the series for ψ k (z) is uniform in every compact subset of D k ∪ ∂D k \ ⋃ M k l=1 {z lk}. This completes the proof of the lemma. It is possible to write ψ k explicitly in the form of a series. But ultimately a primitive of ψ k is needed. In order to properly define it we fix a point w ∈ D\{∞} and introduce the functions (24) ϕ m (z) = and (25) Ω(z) = n∑ m=1 − ∫ z w ∗ (m) ψ m (ζ) dζ + ϕ m (w ∗ (m)), m = 1, 2, . . . , n, ( ) ( ) [ϕ ] m z(m) ∗ − ϕ m w(m) ∗ n∑ (1 − iξ m ) ln a m − z a m − w + 1 2 m=1 n∑ ∑M m m=1 l=1 β lm ln z lm − z z lm − w , where a single valued branch of the logarithm is fixed in such a way that all cuts of ln a m − z a m − w , ln z lm − z z lm − w lie in D ∪ D m ∪ ∂D m and ln x is real for positive x → +∞. In calculating the integral (24), the following relation is used [16] d ( (26) [ϕ m z ∗ dz (m)) ] ( ) 2 rk dϕ ( ) m = − z(m) ∗ , |z − a k | > r k . z − a k dz
12 (2012), No. 2 Schwarz-Christoffel Formula for Multiply Connected Domains 457 The functions Ω(z) and ϕ m (z) belong to ) n∑ ∑M m C A (D, β lm ln(z − z lm ) and to m=1 l=1 ( ) C A D m , 1 ∑M m β lm ln(z − z ml ) , 2 l=1 respectively. One can see from (24) that the function ϕ m (z) is determined by ψ m (z) up to an additive constant which vanishes in (25). The function Ω(z) vanishes at z = w. Integrate each functional equation (19). Application of (24) yields functional equations ϕ k (z) = ∑ ( ) ( ) [ϕ ] m z(m) ∗ − ϕ m w(m) ∗ − ∑ (1 − iξ m ) ln a m − z (27) a m − w m≠k m≠k for the functions + 1 2 n∑ ∑M m m=1 l=1 β lm ln z − z lm w − z lm + c k , |z − a k | ≤ r k , k = 1, . . . , n, ϕ k ∈ C A ( and undetermined constants c k . D k , 1 2 ∑M k l=1 β lk ln(z − z lk ) Lemma 3. The system of functional equations (27) with fixed c k has a unique solution in ( ) C A D k , 1 ∑M k β lk ln(z − z lk ) , k = 1, . . . , n. 2 l=1 This solution can be found by the method of successive approximations. Proof. The proof follows from Lemma 2, since (27) is the result of the integral operator (28) F ↦→ applied to (19). Convergence in C A ( D k , 1 2 ∫ z ∑M k l=1 w ∗ (k) F (t)dt β lk ln(z − z lk ) ) )
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Schwarz-Christoffel Formula for Multiply Connected Domains

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