I was introduced the following example to illustrate the dependence between $z$ and $\bar{z}$, which is the complex conjugate of $z\in \mathbb{C}$:
\begin{equation}
g(z)=\int_{0}^{2 \pi} \frac{d \theta}{2 \pi} \frac{1}{z-e^{i \theta}}=\oint \frac{d w}{2 \pi i} \frac{1}{w(z-w)}
\end{equation}
which appears to only depend on $z$. However, $g\left(z, \bar{z}\right)=\theta\left(|z|^{2}-1\right) / z$ is non-analytic, and depends on $\textbf{both}$ $z$ and $\bar{z}$.
I don't understand this last sentence.
If we know $z$, then we know $|z|^2$ and also $\bar{z}$. All in all, I do not understand why we have to write $g(z,\bar{z})$ as a function of two different variables.
Best Answer
Don't blame yourself for not understanding it. It is not clear at all.
It seems to me that whoever wrote that was aiming at the concept of Wirtinger derivatives: if $U\subset\mathbb C$ and $f$ is a map from $U$ into $\mathbb C$, then, if you see $f(z)$ as $f(x+yi)$ (with $x,y\in\mathbb R$, we define$$\frac{\partial f}{\partial z}=\frac12\left(\frac{\partial f}{\partial x}-i\frac{\partial f}{\partial y}\right)\text{ and }\frac{\partial f}{\partial\overline z}=\frac12\left(\frac{\partial f}{\partial x}+i\frac{\partial f}{\partial y}\right).$$In a sense, $\frac{\partial}{\partial\overline z}$ measures how much $f$ depends on $\overline z$; for instance, $\frac{\partial\overline z}{\partial z}=0$ and $\frac{\partial\overline z}{\partial\overline z}=1$. And $f$ is differentiable at $z_0$ if and only if $\frac{\partial f}{\partial\overline z}(z_0)=0$.