Proposition

Prove that $f(z) = \overline{z}^2$ does not have an antiderivative in any nonempty region.

Solution

Suppose it has an antiderivative $F(z)$. Then $F(z)$ is analytic, so $F’(z) = f(z)$ is analytic.

However, $f(z) = (x^2 - y^2) + i(-2xy)$ does not satisfy the Cauchy-Riemann equations because $u_x = 2x \ne -2x = v_y$ in any nonempty region.