I do not know how helpful my post would be but there is a generalization of the Schwarz reflection principle. First of all if , the conformal map, takes the real line to a straight line in then will map the real axis to the real axis where is the angle the line makes with the positive x-axis. Maybe this realization is all you need in whatever problem you are working on. Second of all there is a generalization. Let be a curve where is an interval. We say that is regular iff is a restriction of some one-to-one analytic function on an open set containing with . If the set happens to be symmetric i.e. (complex conjugation) then we can define "reflection across ". Let , define . We see that points on are invariant under this reflection and that , exactly how we expect a reflection to behave. In general what happens is that if is analytic function on a region that contains a regular curve and that is part of another regular curve , define as reflection of across , and define as reflection of across , then we can extend to by defining for all .In the proof the Schwarz reflection principle is applied to a situation where function is not real valued on the real line segment, but it maps this segment to a straight line in C.
Why can we apply a reflection principle here, which assumes that function must be real valued?