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Schwarz reflection principle (Theorem)

For a region $ G \subset {\mathbb{C}}$ define $ G^* := \{ z : \bar{z} \in G \}$ (where $ \bar{z}$ is the complex conjugate of $ z$). If $ G$ is a symmetric region, that is $ G = G^*$, then we define $ G_+ := \{ z \in G : \operatorname{Im} z > 0 \}$, $ G_- := \{ z \in G : \operatorname{Im} z < 0 \}$ and $ G_0 := \{ z \in G : \operatorname{Im} z = 0 \}$.

Theorem 1   Let $ G \subset {\mathbb{C}}$ be a region such that $ G = G^*$ and suppose that $ f \colon G_+ \cup G_0 \to {\mathbb{C}}$ is a continuous functions that is analytic on $ G_+$ and further that $ f(x)$ is real for $ x \in G_0$ (that is for real $ x$), then there is an analytic function $ g : G \to {\mathbb{C}}$ such that $ g(z) = f(z)$ for $ z \in G_+ \cup G_0$.

That is you can “reflect” an analytic function across the real axis. Note that by composing with various conformal mappings you could generalize the above to reflection across an analytic curve. So loosely stated, the theorem says that if an analytic function is defined in a region with some “nice” boundary and the function behaves “nice” on this boundary, then we can extend the function to a larger domain. Let us make this statement precise with the following generalization.

Theorem 2   Let $ G, \Omega \subset {\mathbb{C}}$ be regions and let $ \gamma$ and $ \omega$ be free analytic boundary arcs in $ \partial G$ and $ \partial \Omega$. Suppose that $ f \colon G \cup \gamma \to {\mathbb{C}}$ is a continuous function that is analytic on $ G$, $ f(G) \subset \Omega$ and $ f(\gamma) \subset \omega$, then for any compact set $ \kappa \subset \gamma$, $ f$ has an analytic continuation to an open set containing $ G \cup \kappa$.

Bibliography

1
John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.
2
John B. Conway. Functions of One Complex Variable II. Springer-Verlag, New York, New York, 1995.



"Schwarz reflection principle" is owned by jirka.
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Other names:  Schwarz reflection theorem, reflection principle
Also defines:  symmetric region
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Cross-references: open set, analytic continuation, compact set, free analytic boundary arcs, domain, function, boundary, analytic curve, reflection, conformal mappings, real axis, real, analytic, continuous functions, complex conjugate, region
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This is version 4 of Schwarz reflection principle, born on 2004-04-13, modified 2005-03-07.
Object id is 5757, canonical name is SchwarzReflectionPrinciple.
Accessed 6080 times total.

Classification:
AMS MSC: 30C35 (Functions of a complex variable :: Geometric function theory :: General theory of conformal mappings)
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