5.5 The Reflection Principle
We have previously considered analytic continuations over two regions with an intersection. Under certain conditions, analytic continuations can be derived across a curve, given by the following theorem.
Theorem 5.5.1 (Painlevé).
Let and be two disjoint simply connected open regions in such that is a simple curve without its endpoints. Let and be two holomorphic functions that are continuous on and , respectively, such that on . Then there exists a unique holomorphic function
on .
Proof.
We aim to prove that the constructed function is holomorphic on . In particular, we only need to prove that is holomorphic on (a neighborhood of) , after which the Identity Theorem (Theorem 3.3.3) applies.
Let be fixed, and choose such that . Let be any simple closed curve in . If is fully contained in , then by Cauchy–Goursat (Theorem 3.1.7),
Similarly, if is fully contained in , then
If intersects , then we can decompose , where is the part of that lies in and is the part of that lies in . The set closes and in the sense that and are both simple closed curves, or unions of simple closed curves (where in each of the two curves have opposite orientations, see Figure 13). By Cauchy–Goursat (Theorem 3.1.7), we have
Hence, by Morera’s Theorem (Theorem 3.2.4), is holomorphic on , and the assertion follows.
A consequent result was discovered by Schwarz, known as the reflection principle, is a unique result derived from the above theorem for when the shared boundary curve lies in the real axis under certain conditions.
Theorem 5.5.2 (Schwarz Reflection Principle).
Let be a connected region on one side of the real axis such that there exists a non-degenerate curve such that . Let be holomorphic with continuity up to such that is real-valued on , and let be the reflection of across the real axis. Then there exists a unique holomorphic function
on .
Proof.
If , then , and since is real on , it follows that for . Thus, we are left to prove that is holomorphic on . Let . It follows that
Since this limit exists, it follows that is holomorphic on . Assume that . Since
it follows that is continuous on . Therefore, by the Painlevé Theorem, is holomorphic on .
This conjugate-symmetry can be generalized by transforming :
Theorem 5.5.3 (Symmetry Principle).
Let be an (infinite) straight line, and let be an open region lying entirely on one side of . Suppose is a non-degenerate open curve contained in . If is holomorphic on , continuous on , and satisfies , where is a straight line, then there exists a unique holomorphic function such that on , where is the reflection of across . Moreover, for any pair symmetric with respect to , the values and are symmetric with respect to .
Proof.
There exist and such that maps to and maps to . Let , which lies entirely on one side of the real axis, and let , a curve on the real axis. The function is holomorphic on and continuous on . By the Schwarz Reflection Principle (Theorem 5.5.2), there exists a unique holomorphic function such that on , where is the reflection of across the real axis. Then is a holomorphic function on such that on . Since linear transformations preserves symmetry, for any pair symmetric with respect to , we have , and thus and are symmetric with respect to . Hence, and are symmetric with respect to .