Schwarz-Christoffel Formula for Multiply Connected Domains

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460 V. Mityushev CMFT Theorem 4. The function (33) yields exp(ω(z)) from the Schwarz-Christoffel integral (7) when all ξ m vanish. The function exp(ω(z)) has the form (34) exp(ω(z)) = F (z; 0, . . . , 0) { ⎡ n∏ ∏M m ( ) ( ) βlm /2 zlm − z n∏ ⎤ zlm − z ∗ βlm /2 (k) = ⎣ ⎦ z m=1 j=1 lm − w z k=1 lm − w(k) ∗ ⎡ ( ) n∏ ⎤ ∏ zlm − z ∗ βlm /2 } (k × ⎣ 1 k) ⎦ · · · z lm − w ∗ k=1 k 1 ≠k (k 1 k) ( n ) ( ∏ n ) a k − w ∏ ∏ a k1 − w(k) ∗ × a k − z a k=1 k=1 k 1 ≠k k1 − z(k) ∗ ( n ) ∏ ∏ ∏ a k2 − w(k ∗ × 1 k) · · · . a k2 − z ∗ k=1 k 1 ≠k k 2 ≠k 1 (k 1 k) Proof. The function F (z; ξ 1 , ξ 2 , . . . , ξ n ) is constructed on the basis of the general solution ψ(z) of the Riemann-Hilbert problem (2) which contains the singularity function (4) as a particular solution. Therefore, F (z; ξ 1 , ξ 2 , . . . , ξ n ) contains the function exp(ω(z)) for appropriate constants ξ 1 , ξ 2 , . . . , ξ n . In order to prove the theorem it is sufficient to check that all singular points of the function (33) coincide to the singular points of the function exp(ω(z)) (including orders of the singularities) if and only if all ξ m are equal to zero. DeLillo et al. [8] used the Schwarz Reflection Principle to prove that the asymptotic behavior of the analytic continuation of exp(ω(z)) near the singular points is described by the relations (35a) (35b) for γ j ∈ E ′ m and (36a) (36b) for γ j ∈ O ′ m. exp(ω(z)) ∼ (z − γ j (z lm )) β lm as z → γ j (z lm ), exp(ω(z)) ∼ (z − γ j (a m )) −2 as z → a m exp(ω(z)) ∼ (z − γ j (z lm )) β lm as z → γ j (z lm ), exp(ω(z)) ∼ (z − γ j (a m )) −2 as z → a m After application of the formula (3) one can observe all these singularities in (8). Though formula (8) does not hold in the general case, asymptotic formulae (35) and (36) always valid for exp(ω(z)) [8]. We now investigate the question, for which parameters ξ 1 , ξ 2 , . . . , ξ n the function F (z; ξ 1 , ξ 2 , . . . , ξ n ) has the same asymptotic (35), (36). Let us fix a m and consider the term µ(z) = z lk − γ e (z) z lk − γ e (w)
12 (2012), No. 2 Schwarz-Christoffel Formula for Multiply Connected Domains 461 from (33) where the last inversion of γ e ∈ E is z(m) ∗ , i.e. γ e(z) can be written as γ e (z) = z(k ∗ pk p−1···k 1 m) with odd p. The relation (11) implies that (37) µ(z) = z − γ t((z lk ) ∗ (m) ) w − γ t ((z lk ) ∗ (m) ) w − γ t (a m ) z − γ t (a m ) , where γ t (z) = z(k ∗ 1 k belongs to 2···k p) O′ m. One can see that (37) gives the required first asymptotic (36) for F (z; ξ 1 , ξ 2 , . . . , ξ n ). It follows from (37) and (3) that the multiplier from (33) ∏M m [µ(z)] β lm/2 contains the multiplier l=1 ∏M m [z − γ t (a m )] −βlm/2 = [z − γ t (a m )] −1 . l=1 Similar arguments can be applied to obtain the required first asymptotic (35) for F (z; ξ 1 , ξ 2 , . . . , ξ n ) by use of ν(z) = z lk − γ o (z) z lk − γ o (w) where γ o ∈ O. One can see also that the multiplier from (33) ∏M m [ν(z)] β lm/2 l=1 contains [z − γ s (a m )] −1 for some γ s ∈ E ′ m. Therefore, the function F (z; ξ 1 , ξ 2 , . . . , ξ n ) contains the multiplier for γ s ∈ E ′ m (the multiplier [z − γ s (a m )] −2+iξm [z − γ t (a m )] −2+iξm for γ t ∈ O ′ m) which determines its behavior near the singular point z = γ s (a m ) (z = γ t (a m )). One can see that F (z; ξ 1 , ξ 2 , . . . , ξ n ) has the same behavior as exp(ω(z)) at these points if and only if ξ m = 0. The theorem is proved. Remark 1. It follows from Theorem 4 that S(z) = O(|z| −3 ) (see (3) and (15)–(16)) that corresponds to [8]. as z → ∞
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Schwarz-Christoffel Formula for Multiply Connected Domains

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