I had come across one proof of Chain rule given by Mark Viola Sir as The complex version of the chain rule
I know that Re(Z)=$\frac{z+\bar z}{2}$ and $Im(z)=\frac{z-\bar z}{2i}$
By that argument we can write $g(f(x,y)$ in given form.
But then Sir had taken derivative by taking Psedu term w , But I do not understand Why Last term Become derivative with respective to z.
Any Help will be appreciated.
Wirtinger derivatives ${\partial\over\partial z}$, ${\partial\over\partial\bar z}$ are around since 1926. Some people even refer to Poincaré (1899). But until today there is no commonly accepted paradigm and way of explaining what they should mean.
A proof of the chain rule in a multivariate evironment would encompass some limiting argument. I don't see one here. This means that the author takes the chain rule for maps $f:\>{\mathbb R}^n\to{\mathbb R}^m$ for granted. His fiddling around on the shown page then is only meant to be an exercise in linear algebra.
In my opinion you can talk at the same time about ${\partial f\over\partial z}$ and ${\partial f\over\partial\bar z}$ when $f$ is a function defined for $z=x+iy$, but you cannot talk at the same time about ${\partial g\over\partial w}$ and ${\partial g\over\partial z}$ when $w=f(z,\bar z)$. To sum it up: I cannot take the shown page seriously.
