Intuition behind the notation of differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline z}$.

Question

I am a graduate student of Mathematics.In this semester I am studying complex analysis.Stein Shakarchi's complex analysis book defines differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline z}$ as follows:

$\frac{\partial}{\partial z}=\frac{1}{2}(\frac{\partial}{\partial x}+\frac{1}{i}\frac{\partial}{\partial y})$ and $\frac{\partial}{\partial \overline z}=\frac{1}{2}(\frac{\partial}{\partial x}-\frac{1}{i}\frac{\partial}{\partial y})$

I want to know the motivation behind denoting them by the partial derivative notation with respect to $z$ and $\overline z$.Is there any reason behind choosing such a convention?

Answer 1

If $z = x + y i$ with $x$ and $y$ real, $\overline{z} = x - y i$. These partial derivatives are just what you get by regarding this as a "change of variables" and using the chain rule.