The Wirtinger derivatives ${\mathrm{\partial }}_{z}f$ and ${\mathrm{\partial }}_{\bar{z}}f$, and antiholomorphic functions

Let $U\subseteq \mathbb{C}$ be open and write a map $f:U\to \mathbb{C}$ as $f=u+iv$ in Euclidean coordinates. At a point where the four real partial derivatives exist, define

${\mathrm{\partial }}_{z}f:=\frac{1}{2}({\mathrm{\partial }}_{x}f-i{\mathrm{\partial }}_{y}f)=\frac{1}{2}(u_x+v_y)+\frac{i}{2}(v_x-u_y),$

${\mathrm{\partial }}_{\bar{z}}f:=\frac{1}{2}({\mathrm{\partial }}_{x}f+i{\mathrm{\partial }}_{y}f)=\frac{1}{2}(u_x-v_y)+\frac{i}{2}(v_x+u_y)\mathrm{.}$

The partial derivatives are those of Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$. If $f$ is real totally differentiable at the point, direct expansion gives the differential identity

$Df(h)=({\mathrm{\partial }}_{z}f)h+({\mathrm{\partial }}_{\bar{z}}f)\bar{h}\mathrm{.}$

A real-differentiable map is antiholomorphic on $U$ when ${\mathrm{\partial }}_{z}f=0$ at every point of $U$. Thus its differential is conjugate-linear at every point; no complex differentiability is asserted unless the other Wirtinger derivative also vanishes.