My question concerns the use of the Schwarz Reflection principle (or principle of symmetry) to reflect the regions of the domain of a holomorphic function in a symmetric region (with respect to a straight line segment) in which $ f (z) $ is already defined.

More specifically, if $ f: mathbb {D} rightarrow mathbb {C} $ is a holomorphic function that has a real value along the line $ arg z = pi / 3 $we can take a "side" of a region whose boundary is along $ arg z = pi / 3 $ and think through it via the formula $ tilde {f} ( tilde {z}) $, where ~ denotes the reflection. As it is an analytical continuation and $ f (z) $ is already defined on the other "side" of $ arg z = pi / 3 $, does it make sense to conclude that $ f (z) = f ( tilde z) $?

Moreover, when the line segment in question is the real axis, the map is actually the usual complex conjugation. Initially, I wanted to tackle this issue by turning $ arg z = pi / 3 $ clockwise $ pi / $ 3 to bring the real value segment back to the axis, apply the principle of symmetry here and rotate backwards. Unless I'm mistaken, I think it shows that $ tilde f ( tilde z) = overline {f ( overline {z})} $ ? That is, reflection by line segments that are not on the real axis always follows the same reflection formula. It seems wrong. I would like to have a glimpse of my two (apparently fake) observations .. thanks!