Let $G \subseteq C$ be an open domain which is contained in some bounded disk, say $D_{0}$ . Further suppose that $δ G$ intersects the boundary of this disk. Now let $f : G \to C$ be holomorphic with a continuous extension $f : c l (G) \to C$ . Further suppose that $f (δ G)$ intersects the boundary of another disk, say $D_{1}$ . We define reflections over these disks in the following way:

σ_{0} : D_{0} - {c e n t e r} \to (c l (D_{0}))^{c} σ_{0} (z) = \frac{r^{2}}{| z - c e n t e r |} (z - c e n t e r) + c e n t e r

$σ_{1}$ can be defined in a similar way. With these conventions, we define:
:?:

F (z) = {\begin{cases} f (z) & z \in c l (G) \\ σ_{1} (f (σ_{0}^{- 1} (z))) & z \in σ_{0} (G) \end{cases}

> [!remark] > We can prove this by pre and post composing with a Möbius transformation, so that we can apply the normal Schwarz reflection.