State the standard Schwarz reflection principle involving reflection across the real axis.
Give a linear fractional transformation \(T\) mapping \(\DD\) to \(\HH\). Let \(g(z) = \bar z\), and show \[ (T^{-1} \circ g \circ T)(z) = 1/\bar z .\]
Suppose that \(f\) is holomorphic on \(\DD\), continuous on \(\bar \DD\), and real on \(S^1\). Show that \(f\) must be constant.
Solution
Goal: (1) State the Schwarz reflection principle across the real axis; (2) give a Möbius map \(T: \DD \to \HH\) and verify \(T^{-1}\circ g \circ T = 1/\bar z\) for \(g(z) = \bar z\); (3) show \(f\) holomorphic on \(\DD\), continuous on \(\bar\DD\), real on \(S^1\), must be constant.
1. Statement of the Schwarz reflection principle (real axis).
Let \(\Omega \subseteq \HH\) be a domain whose boundary contains an open interval \(I\) of the real axis, and let \(f\) be holomorphic on \(\Omega\), continuous on \(\Omega \cup I\), with \(\Im f(z) \to 0\) as \(z \to I\) (equivalently \(f\) real on \(I\)). Then \(f\) extends holomorphically across \(I\) to \(\Omega \cup I \cup \overline\Omega\) by \(f(\bar z) = \overline{f(z)}\).
2. Give \(T: \DD \to \HH\).
\(T(z) = i\frac{1 + z}{1 - z}\) maps the unit disk onto the upper half-plane (the Cayley transform): \(z \in \DD\) iff \(\Im T(z) > 0\). Its inverse is \(T^{-1}(w) = \frac{w - i}{w + i}\).
3. Verify \(T^{-1} \circ g \circ T = 1/\bar z\) with \(g(z) = \bar z\).
Compute: \(g(T(z)) = \overline{T(z)} = \overline{i\frac{1+z}{1-z}} = -i\frac{1+\bar z}{1 - \bar z}\). Then \(T^{-1}(\overline{T(z)}) = \frac{\overline{T(z)} - i}{\overline{T(z)} + i} = \frac{-i\frac{1+\bar z}{1-\bar z} - i}{-i\frac{1+\bar z}{1-\bar z} + i} = \frac{-i(1+\bar z) - i(1 - \bar z)}{-i(1+\bar z) + i(1-\bar z)} = \frac{-2i}{-2i\bar z} = \frac{1}{\bar z}\).
4. Setup for (3): \(F = f \circ T^{-1}\) is holomorphic on \(\HH\), continuous on \(\bar\HH\), and real on \(\RR\); extend by reflection to an entire function.
\(T^{-1}\) maps \(\HH\) onto \(\DD\) and \(\RR\) onto \(S^1 \setminus \{1\}\); since \(f\) is real on \(S^1\), \(F\) is real on \(\RR\) (minus possibly one point, removable). By the reflection principle (<1>1), \(F\) extends holomorphically to all of \(\CC\) via \(F(\bar w) = \overline{F(w)}\). The function \(f\) is bounded on the compact \(\bar\DD\) (continuous), so \(F\) is bounded on \(\bar\HH\); the reflected extension is bounded on the lower half-plane too, hence \(F\) is a bounded entire function. By Liouville, \(F\) is constant.
5. Q.E.D.
4: \(F\) constant ⟹ \(f\) constant. All three parts done: 1 states the principle, <1>2–<1>3 handle the Cayley transform computation, <1>4–<1>5 prove constancy.