How to prove $\frac{\partial z}{\partial \bar{z}} = 0$ if and only if $\frac{\partial \bar{z}}{\partial z} = 0$

Question

Here is a problem form my complex analysis HW.

Unfortunately, I really have no idea how to go about this. Specifically, I don't really know how to take partials of that form. Does anyone have anything that might help me?

Answer 1

Hint: If $z=x+iy$, usually you define: $$\frac{\partial}{\partial z}=\frac{1}{2}\left(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y}\right)$$ $$\frac{\partial}{\partial \bar{z}}=\frac{1}{2}\left(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y}\right)$$ Can you take it from here?

Answer 2

You can compute $\mathrm{d}z$ and $\mathrm{d}\bar{z}$ in terms of the basis $\mathrm{d}x$ and $\mathrm{d}y$.

Then you can compute $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \bar{z}}$ in terms of $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$.

Combining $\mathrm{d}z$ with $\frac{\partial}{\partial \bar{z}}$ gives $\frac{\partial z}{\partial \bar{z}}$ (using, for example, that combining $\mathrm{d}x$ with $\frac{\partial}{\partial x}$ gives $1$), and similarly for the other one.

I don't think the problem, or the suggested method, actually make sense. If $(z, \bar{z})$ truly to be interpreted as a "coordinate system", then the values of $\frac{\partial z}{\partial \bar{z}}$ and the other three combinations would follow from the usual conventions of what partial differentiation notation actually means (i.e. "take the derivative in the direction where this 'coordinate' increases and the other one is held constant"), and have nothing to do with whether or not $z$ is a "holomorphic coordinate".