A Question from complex variable

Question

Show that an analytic function with constant modulus is itself a constant

Answer 1

Hint: Write $f = u+iv$. If $|f| = 0$ then $f = 0$ and there's nothing to do. Otherwise, $u²+v² = c²$ for some constant $c > 0$. Differentiate implicitly w.r.t. $x$ and $y$ and use the Cauchy-Riemann equations to conclude that $u_x = u_y = v_x = v_y \equiv 0$.