Schwarz-Christoffel Formula for Multiply Connected Domains

454 V. Mityushev CMFT
Introduce the function
(15) Φ(z) =
where
(16) Φ k (z) = 1 2
n∑
Φ k (z),
k=1
∑M k
l=1
β lk
z − z lk
.
Then ψ(z) ∈ C A (D, 2Φ) and ψ k (z) ∈ C A (D k , Φ k ). The problems (2) and (13) in
the classes considered are equivalent in the sense of the following result.
Lemma 1.
(i) If ψ(z) and ψ k (z) are solutions of (13) in the class considered, then ψ(z)
satisfies (2).
(ii) If ψ(z) is a solution of (2), there exist functions ψ k ∈ C A (D k , Φ k ) and real
constants ξ k such that the R-linear conditions (13) are fulfilled.
Proof. The proof of the first assertion is evident. It is sufficient to take the real
part of (13).
Conversely, let ψ(z) satisfy (2). The function
Ψ k (z) = iξ k
2 + (z − a k)ψ k (z)
can be uniquely determined from the simple Schwarz problem for the disk D k
[10, 16]
(17) 2 Im Ψ k (t) = Im(t − a k )ψ(t), |t − a k | = r k .
It is assumed that the function Ψ k (z) is continuous in |z − a k | ≤ r k except at
the points z lk , where the principal part β lk (z lk − a k )/[2(z − z lk )] of Ψ k (z) is
determined by the right hand part of (17). The problem (17) for the function
Ψ k (z) has a unique solution, since Re Ψ k (a k ) = 0. Therefore, the function
ψ k (z) and the constant ξ k are uniquely determined in terms of ψ(z) for each
k = 1, . . . , n. Direct calculations yields the asymptotic (14).
Hence the lemma is proved.
We now proceed to solve the R-linear problem (13) written in the form
( ) 2 rk
(18) ψ(t) = ψ k (t) − ψ k (t) − 1 − iξ k
, |t − a k | = r k , k = 1, . . . , n.
t − a k t − a k