complex differentiation, alternative way?

Question

$\partial/\partial\bar{z}$ is defined as $1/2[\partial/\partial x+i\partial/\partial y]$. So lets say you have a function $f(z,\bar{z})$ in order to find $\partial f/\partial \bar{z}$ I have to write f as $f(x+iy,x-iy)$ and calculate $1/2[\partial/\partial x+i\partial/\partial y]f(x+iy,x-iy)$. But I am wondering, can I instead differentiate naivly in terms of $\bar{z}$ on $f(z,\bar{z}) $as I do in real partial-differentiation?

Answer 1

Yes. if you use the Hj0rungnes' notation (see the book Complex-valued matrix derivatives), then you will have

$$ \frac{\partial f}{\partial z} = \frac{1}{2}\cdot (\frac{\partial f}{\partial x} - i \frac{\partial f}{\partial y}),$$ and

$$ \frac{\partial f}{\partial \bar{z}} = \frac{1}{2}\cdot (\frac{\partial f}{\partial x} + i \frac{\partial f}{\partial y}).$$

With this notation, $\bar{z}$ and $z$ are treated as independent variables.

Answer 2

Yes. It's a handy way to quickly check whether or a complex function is differentiable or holomorphic.

For example the modus squared function $f(z) = |z|^2 = z\bar{z}$. As $\displaystyle \frac{\partial f}{\partial \bar{z}} = z \neq 0$, $f$ is not holomorphic.