I was wondering if complex conjugation "commutes" with differentiation in the sense that
$$ \overline{\partial_z f(z,\bar z)} = \partial_{\bar z} f(\bar z,z)$$
where $f(z,\bar z)$ is a well-behaved function. My intuition is that this is true because the left hand side is the complex conjugate of $\frac{f(z+h,\bar z) – f(z,\bar z)}{h}$ in the limit $h \to 0$, and complex conjugation commutes with these operations. I would appreciate a more definite answer, or a different relation between the two expressions.
Best Answer
No. For $f(z) = i z$, $\overline{\partial_z f(z, \bar z)} = -i$ but $\partial_{\bar{z}} f(\bar z, z) = i$.
In general, you have the property of the complex partial derivatives that $\overline{\partial_z f} = \partial_{\bar{z}} \bar{f}$, so you are really asking whether $f(\bar z) = \overline{f(z)}$. This is the case when $f$ is a holomorphic function whose restriction to the real numbers is real-valued.