What is relationship between Wirtinger differential operator and multivariable chain rule?

Question

What is relationship between Wirtinger differential operator(equation 5) and multivarible chain rule(equation 4)? for other Wirtinger related questions look here.

Answer 1

Let $\Omega$ be an open set in $\mathbf{C}$ (which identify with $\mathbf{R}^2$) and $f\in C^1(\Omega)$; as $2dx=dz+d\bar{z}$ and $2idy=dz-d\bar{z}$ we have $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy=\frac{\partial f}{\partial z}dz+\frac{\partial f}{\partial\bar{z}}d\bar{z},$$ defining as usual $$\frac{\partial f}{\partial z}:=\frac{1}{2}\frac{\partial f}{\partial x}+\frac{1}{2i}\frac{\partial f}{\partial y}, \ \ \ \mathrm{and} \ \ \ \frac{\partial f}{\partial \bar{z}}:=\frac{1}{2}\frac{\partial f}{\partial x}-\frac{1}{2i}\frac{\partial f}{\partial y}.$$ Equation (4) is merely formal ($f$ is considered as a function of $z$ and $\bar{z}$, which are treated as independent variables even though they are conjugate); the notation is very suggestive, however if $f$ is holomorphic (so that $\partial f/\partial\bar{z}=0$) we have in fact $df/dz=\partial f/\partial z$ (where the left hand side is the complex derivative, see for instance Hörmander's An introduction to complex analysis in several variables.