On conformal mapping of polygonal regions (Q1921739)

Let \(\overline C\) denote a closed plane and \(H^+\) its upper half-plane. Let \(D_n\) stand for the interior of an \(n\)-gon whose internal angles at vertices \(A_k\) are equal to \(\alpha_k\pi\), \(k=1, \dots, n\). It is known that, for any polygon, the relation \(\sum^n_{k=1} \alpha_k=n-2\) holds. The following theorem is a central result in the theory of conformal mappings of polygonal regions [\textit{R. Schinzinger} and \textit{P. Laura}, Conformal mapping: methods and applications, Elsevier, Amsterdam (1991), p. 65]. Theorem, Suppose that \(D_n\) is a simply connected domain in the complex plane \(\mathbb{C}\), bounded by a polygon with vertices at points \(A_1, \dots, A_n\) and internal angles \(\pi \alpha_k\) where \(0\leq \alpha_k\leq 2\) for finite \(A_k\) and \(-2\leq \alpha_k\leq 0\) for \(A_k=\infty\). Then there exists a conformal mapping of the upper half-plane \(H^+\) onto \(D_n\) and, moreover, each mapping of this sort can be represented in the form \[ f(z)=c\int^z_0 \prod^n_{k=1} (z-a_k)^{\alpha_k-1} dz+c_1 \] where \(a_1, \dots, a_n\) are the preimages of the vertices \(A_1, \dots, A_n\). The complex constants \(a_1, \dots, a_n\), \(c\) and \(c_1\) are called the assessory parameters of the Schwarz-Christoffel integral. The principal problem is to determine these parameteter. The authors, developing \textit{P. P. Kufarev}'s method [Dokl. Akad. Nauk SSSR, N. Ser. 57, 535-537 (1947; Zbl 0041.24302)], solve this problem.