Show that a real-valued holomorphic function must be constant.
Solution
Goal: Show that a real-valued holomorphic function on a domain must be constant.
1. Write \(f = u + iv\) with \(v \equiv 0\); the Cauchy–Riemann equations give \(u_x = v_y = 0\) and \(u_y = -v_x = 0\).
If \(f\) is holomorphic, the Cauchy–Riemann equations \(u_x = v_y\), \(u_y = -v_x\) hold. Since \(f\) is real-valued, \(v \equiv 0\), so \(v_x = v_y = 0\); hence \(u_x = u_y = 0\) on the domain.
2. \(u\) is constant on the domain.
Step 1 shows both partial derivatives of \(u\) vanish. On a connected (path-connected) domain, fix \(z_0\); any other point joins to \(z_0\) by a path, and along the path \(u\) is constant (vanishing directional derivative), so \(u(z) = u(z_0)\) everywhere.
3. QED \(f = u\) (as \(v \equiv 0\)) is constant by step 2.