Calculating a complex derivative of a polynomial What are the rules for derivatives with respect to $z$ and $\bar{z}$ in polynomials?
For instance, is it justified to calculate the partial derivatives of $f(z,\bar{z})=z^3-2z+\bar{z}-(\overline{z-3i})^4$ as if $z$ and $\bar{z}$ were independent? i.e. $f_z=3z^2-2$ and $f_\bar{z}=1-4(\overline{z-3i})^3$
 A: Still, for $f(z)=\bar{z}$ we have $f_z=0$, $f_\bar{z}=1$ (by definiton), which is consistent with the "rules" above. So - are they correct? ($f$ is harmonic, not analytic)
A: I would first write $$ f(z,\bar z)=z^3−2z+\bar z−(\bar z+3i)^4 $$
and then treat $z$ and $\bar z$ as independent parameters.
Then I have
$$f_z=3z^2−2$$      $$f_{\bar z}=1−4(\bar z+3i)^3$$
Am I right?
A: This is addressed in this answer. We have
$$
\frac{\partial}{\partial z}\bar{z}=0\quad\text{and}\quad\frac{\partial}{\partial\bar{z}}z=0
$$
and thus the chain rule for partial derivatives says we can treat $z$ as constant when applying $\dfrac{\partial}{\partial\bar{z}}$ and treat $\bar{z}$ as constant when applying $\dfrac{\partial}{\partial z}$. Therefore,
$$
\frac{\partial}{\partial z}\left(z^3-2z+\bar{z}-(\overline{z-3i})^4\right)=3z^2-2
$$
and
$$
\frac{\partial}{\partial\bar{z}}\left(z^3-2z+\bar{z}-(\overline{z-3i})^4\right)=1-4(\overline{z-3i})^3
$$
A: no, for instance $f(z) = \overline z$ is not differentiable at any $z_0 \in \mathbb{C}$
