Maybe I understand it wrong. I don't feel relax if I don't ask this question.
I study complex analysis myself.
In the nptel's video(given below), Prof. V. Balakrishnan says that If a function of $z=x+iy$ is not function of $\bar z=x-iy$ then it is an analytical function. Moreover, at the beginning of the video he mentions about linear independence of $z=x+iy$ and $\bar z=x-iy$. And at about 22th minute of the video he writes $\dfrac{\partial f}{\partial \bar z}=0$ and he concludes the Cauchy-Riemann conditions for a function to be differentiable.
Question: 1) What is relationship between linearly independence and derivation? I mean how generally can we talk about differentiable of something with respect to something. In this example we can directly say the below(question 2) that to be zero. I want to understand when I can do it.
2) What come that $\dfrac{\partial f}{\partial \bar z}=0$ I attempted to show this by using chain rule but I couldnot finish this $$\dfrac{\partial f}{\partial \bar z}=\dfrac{\partial f}{\partial x}\dfrac{\partial x}{\partial (x-iy)}+\dfrac{\partial f}{\partial y}\dfrac{\partial y}{\partial (x-iy)}$$
Note: He uses $z^*=x-iy$ instead of $\bar z=x-iy$
The video:https://youtu.be/b5VUnapu-qs?t=21m32s
By definition, the Wirtinger derivative is $$ {\partial\over\partial\overline{z}}=\frac12\left({\partial\over\partial x}+i{\partial\over\partial y}\right) $$ This is just a formal definition of a differential operator. If $f$ is an analytic function, you can check that the condition $$ {\partial f\over\partial\overline{z}}=0$$ is equivalent to the Cauchy-Riemann equations.
Similarly, with the formal definition $$ {\partial\over\partial z}=\frac12\left({\partial\over\partial x}-i{\partial\over\partial y}\right)$$
you can check that $$ {\partial f\over\partial z}=f'(z)$$
This is sometimes expressed by saying that and analytic function $f$ is a function of $z$ alone, not of $\overline{z},$ but this is just an informal statement. Obviously, $z$ and $\overline{z}$ are not independent variables, and it makes no real sense to talk about partial derivatives with respect to them.