Complex conjugate differential equation

Question

I am curious how the conjugate function is defined when the function itself is a differential equation. For example:

$f(z) = \frac{dg(z)}{dz} + \frac{d^2g(z)}{dz^2}$

How then is $\bar{f}(z)$ defined?

$\bar{f}(z) = \frac{d\bar{g}(z)}{dz} + \frac{d^2\bar{g}(z)}{dz^2}$ ?

$\bar{f}(z) = \frac{d\bar{g}(z)}{d\bar{z}} + \frac{d^2\bar{g}(z)}{d\bar{z}^2}$ ?

Something else? To be clear, I'm a novice in this field and have only ever taken introductory courses in functional analysis, so an obvious answer may not seem so obvious to me.

Thanks!

Answer 1

The second one. In two words, conjugate everything. As an analogy, the conjugate of $g/z$ is $\bar{g}/\bar{z}$.