Consider $u(z)=\ln(|z|^2)=\ln(x^2+y^2)$. I know that $u$ does not have a harmonic conjugate from $\mathbb{C}\setminus\{0\}\to\mathbb{R}$ but playing around with partial derivatives and integrating around the unit circle.
However, I know that a function $u$ has a harmonic conjugate if and only if its conjugate differential $*du$ is exact. This is defined as $*du=-\frac{\partial u}{\partial y}dx+\frac{\partial u}{\partial x}dy$.
I calculate this to be $ *du=\frac{-2y}{x^2+y^2}dx+\frac{2x}{x^2+y^2}dy $ so I would assume this is not exact. Is there a way to see that easily? Is this how the criterion for existence or nonexistence of a harmonic conjugate is usually applied in terms of the conjugate differential? Thanks.