On analytic continuation of conformal mapping of a circular triangle

Given a circular triangle $T$ having a zero angle at a point at infinity and two equal nonzero angles, it is shown that a conformal mapping of $T$ onto a half-plane can be continued to a semi-infinite strip by applying the Riemann–Schwarz symmetry principle. The problem of analytic continuation of such a mapping arises as an auxiliary problem in constructing a conformal mapping of an $L$-shaped domain onto a half-plane.