Let $u(z) = \text{Im}(1/z^2)$ and set $u(0) = 0$
We have that $u(z) = \text{Im}({\bar{z}^2/|z|^4)}$ and still that $u(0) = 0$
We then have that if $z \neq 0$ then $$u(z) = \frac{-2xy}{(x^2+y^2)^2} $$
I want to show that all the partial derivatives of $u$ with respect to $x$ and respect to $y$ exist for all $z \in \mathbb{C}$ and that $\Delta u = 0$. But nonethless conclude that $u(z)$ is not Harmonic.
I'm confused on the first step because it does not appear that any partial derivative with respect to $x$ or $y$ exist for all $z \in \mathbb{C}$ as this function not even continuous at $0$. I think this is easiest to see if you convert to polar coordinates and let $r \to 0$. So should the problem just be to show the aforementioned derivatives existing for all $z \in \mathbb{C} / \{0\}$ and that $\Delta u = 0$ for all $z \neq 0$ ?
It's more likely that I'm making a mistake than that my textbook is, so can someone enlighten me?