10.5 The Schwarz Reflection Principle
The next result is the most useful concrete continuation theorem in the subject: it continues a function across a segment of the real axis, and it does so by an explicit formula.
Theorem 10.13 (Schwarz Reflection Principle). Let \(D\) be a domain symmetric about the real axis, that is, \(z \in D\) if and only if \(\overline {z} \in D\). Write \[D^{+} = D \cap \{\operatorname {Im}(z) > 0\},\quad D^{-} = D \cap \{\operatorname {Im}(z) < 0\},\quad D^{0} = D \cap \mathbb {R}.\] Suppose \(f\) is analytic on \(D^{+}\), continuous on \(D^{+} \cup D^{0}\), and real-valued on \(D^{0}\). Then \(f\) continues analytically to the whole of \(D\), and the continuation is given by \[f(z) = \overline {f(\overline {z})}\quad \text {for } z \in D^{-}.\]
Proof. Define \(g(z) = \overline {f(\overline {z})}\) on \(D^{-}\). That \(g\) is analytic is a direct check on the Cauchy–Riemann equations: writing \(f = u + iv\), the function \(g\) has real part \(u(x,-y)\) and imaginary part \(-v(x,-y)\), and the two pairs of partial derivatives each pick up a compensating sign. On \(D^{0}\), \(f\) is real, so \(g\) and \(f\) agree there and the function built from \(f\) on \(D^{+}\), \(g\) on \(D^{-}\) and the common values on \(D^{0}\) is continuous on \(D\). Analyticity across the segment then follows from Morera’s theorem: the integral of the combined function around any triangle in \(D\) is zero, since a triangle meeting \(D^0\) may be split into pieces lying in the closed half-planes and the contributions along the real segment cancel. \(\blacksquare \)
The hypothesis that \(f\) be real on \(D^0\) is essential, and it is what makes \(\overline {f(\overline {z})}\) the right formula: reflection in the real axis on the \(z\)-side must be matched by reflection in the real axis on the \(w\)-side. □
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