Complex Analysis: Theory and Applications · Problem 3.2.15

The key insight is to use the Cauchy–Riemann equations, which are the necessary condition for analyticity. If $f = u + iv$ is analytic, then $u$ and $v$ satisfy specific partial derivative relationships. When you take the complex conjugate to form $g = \overline{f} = u - iv$, those relationships flip sign—and since $f$ is nonconstant, you can show that $g$ cannot simultaneously satisfy the Cauchy–Riemann equations. This contradiction reveals that conjugation destroys analyticity.

The key insight is to use the Cauchy–Riemann equations, which are the necessary condition for analyticity. If $f = u + iv$ is analytic, then $u$ and $v$ satisfy specific partial derivative relationships. When you take the complex conjugate to form $g = \overline{f} = u - iv$, those relationships flip sign—and since $f$ is nonconstant, you can show that $g$ cannot simultaneously satisfy the Cauchy–Riemann equations. This contradiction reveals that conjugation destroys analyticity.