TITLE: Why $\partial$ and $\bar{\partial}$ defined in that way (the Wirtinger derivatives)?
QUESTION [11 upvotes]: For $\mathbb{C}$-valued functions, why are $\frac{\partial}{\partial z}$ and  $\frac{\partial}{\partial \bar{z}}$ defined as
$$
  \frac{\partial}{\partial z}=
  \frac{1}{2}\left(
    \frac{\partial}{\partial x} - i
    \frac{\partial}{\partial y}
  \right)
  \quad \quad
  \frac{\partial}{\partial \bar{z}}=
  \frac{1}{2}\left(
    \frac{\partial}{\partial x} + i
    \frac{\partial}{\partial y}
  \right)
$$
where $x$ and $y$ are the real and imaginary parts, respectively.
It seems to be me that the inventor of this notation could have easily reverse the signs: Let $\frac{\partial}{\partial z}$ have a $+$ in the middle and let $\frac{\partial}{\partial \bar{z}}$ have a $-$ in the middle instead.  Is there any reason that they are defined the way they are now? The same question extends to the $\partial$ and $\bar{\partial}$ operators.
One possible guess is that in complex analysis, one usually works with holomorphic functions, so one operator is used much more often than the other. We probably want the more frequently used one to be the one that is easier to write. But I really doubt this is the reason. After all, it's just one extra stroke.

REPLY [6 votes]: By the chain rule,
$$
\frac{\partial}{\partial z} = \frac{\partial}{\partial x} \cdot \frac{\partial x}{\partial z} + \frac{\partial}{\partial y} \cdot \frac{\partial y}{\partial z}. 
$$
Now
$$\frac{\partial x}{\partial z} = \frac{\partial \frac{z + \bar{z}}{2}}{\partial z} = \frac{1}{2}
\quad \text{and} \quad \frac{\partial y}{\partial z} = \frac{\partial \frac{z - \bar{z}}{2 i}}{\partial z} = \frac{1}{2i} = - \frac{i}{2}.$$
Therefore
$$
\frac{\partial}{\partial z} = \frac{1}{2}\frac{\partial}{\partial x}  - \frac{i}{2} \frac{\partial}{\partial y}. 
$$
Similar derivation gives the formula for $\frac{\partial}{\partial \bar{z}}$.