TITLE: Why can we view $z$ and $\bar z$ as independent variables in complex analysis?
QUESTION [18 upvotes]: I am quite confused about how to understand $\frac{\partial f}{\partial z}f(z,\bar z).$ Do $z$ and $\bar z$ in $f(z,\bar z)$ act the same way as $x$ and $y$ in $f(x,y)$? 
If so, how can we prove this?

REPLY [6 votes]: All this makes perfect sense; you just have to use the right definitions:
Let $f: \mathbb{C}\simeq \mathbb{R}^2\to \mathbb{C}$ be real analytic, that is, if you think of it as being defined on $\mathbb{R}^2$ it has a power series expansion in $(x,y)$ at every point in $\mathbb{R}^2$ (note that this is weaker than being holomorphic). Then $f$ can be extended to a holomorphic function $D\to\mathbb{C}$, where $D\subset \mathbb{C}^2$ is a neighborhood of $\mathbb{R}^2 \subset \mathbb{C^2}$, so you can write $f(x,y)$ where $x,y$ are now complex! Now consider the following change of variables:
$$(z,\bar z)=(x+iy,x-iy)$$
Here we regard $\bar z$ as just a symbol that is independent of $z$. Then it makes sense to write $f(z,\bar z)$; $\partial_z$, $\partial_{\bar z}$ are the Wirtinger derivatives, and $\partial_{\bar z}f=0$ (that is, $f$ is actually independent of $\bar z$) is equivalent to $f$ being holomorphic.