Conformal mappings of a half-plane onto domains with transfer symmetry (Q1589055)

Let \(D\) be a simply connected domain in the complex plane and suppose that the boundary of \(D\) consists of segments and rays and has transfer symmetry; the latter condition means that there exists a positive constant \(T\) such that if \(w\) is a boundary point, then \(w+T\) is also a boundary point. Assume also that the boundary arc from \(w\) to \(w+T\) consists of a finite number of segments or rays. The case of domains with \(T\)-transfer symmetry for all \(T\) (e.g. a half-plane) is excluded. Under these conditions, the author offers a Schwarz-Christoffel-type formula for the conformal mapping of the upper half-plane onto \(D\). He also provides some examples where the formula is applied.