Wirtinger derivatives and conjugate

Question

I haven't found anywhere in the literature (that's available to me, at least) a proper explanation of the following relations for a function $f \in \mathcal{C}(\Omega)$, $\Omega$ domain of $\mathbb{C}^n$ $$\frac{\overline{\partial f}}{\partial{z_i}} = \frac{\partial \overline{f}}{\partial\overline{z}_{i}}$$ $$ \frac{\overline{\partial f}}{\partial\overline{z}_{i}} = \frac{\partial \overline{f}}{\partial z_{i}}$$ Why are they (must they be) true?

Answer 1

To avoid indices, work with one complex variable $z$. Write down the definition of Wirtinger derivatives: $$ f_z = \frac12(f_x-if_y),\qquad f_{\bar z} = \frac12(f_x+if_y) \tag1$$ Apply complex conjugate to (1), observing by the way that $\overline{f_x}=(\bar f)_x$, etc. $$ \overline{f_z} = \frac12(\bar f_x+i\bar f_y),\qquad \overline{f_{\bar z}} = \frac12(\bar f_x-i\bar f_y) \tag2$$ In Wirtinger notation, (2) takes the form $$ \overline{f_z} = (\bar f)_{\bar z},\qquad \overline{f_{\bar z}} = (\bar f)_z \tag3$$ as claimed.